![Page 1: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/1.jpg)
Outline
Robust Optimization: Theory and Tools
Marcela Mera Trujillo1
1Math DepartmentWest Virginia UniversityMorgantown, WV USA
April 28, 2015
M. Mera Trujillo Optimization Methods in Finance
![Page 2: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/2.jpg)
Outline
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)
2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 3: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/3.jpg)
Outline
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 4: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/4.jpg)
Outline
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 5: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/5.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 6: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/6.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 7: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/7.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 8: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/8.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:
* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 9: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/9.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.
* There are constraints with uncertain parameters that must be satisfied regardless of thevalues of the these parameters.
* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 10: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/10.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied
regardless of thevalues of the these parameters.
* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 11: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/11.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.
* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 12: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/12.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function
or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 13: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/13.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions
are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 14: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/14.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions are sensitive to perturbations.
* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 15: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/15.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Generalities
Robust Optimization refers to the modeling of optimization problems with datauncertainly.
Robust optimization models can be useful in the following situations:* Some of the problem parameters are estimates and carry estimation risk.* There are constraints with uncertain parameters that must be satisfied regardless of the
values of the these parameters.* The objective function or the optimal solutions are sensitive to perturbations.* The decision-maker cannot afford to low-probability but high-magnitude risks.
M. Mera Trujillo Optimization Methods in Finance
![Page 16: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/16.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 17: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/17.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?
A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 18: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/18.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.
Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 19: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/19.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.
Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 20: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/20.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters
(if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 21: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/21.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).
Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 22: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/22.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative
(worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 23: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/23.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 24: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/24.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:
Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 25: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/25.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.
There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 26: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/26.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.
The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 27: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/27.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.
The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 28: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/28.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks
(typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 29: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/29.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Robust Optimization (RO))
what is it?A complementary methodology to stochastic programming and sensitivity analysis.Seeks a solution that will have an “acceptable” performance under most realizations ofthe uncertain inputs.Usually, no distribution assumption is made on uncertain parameters (if such informationis available, it can be utilized beneficially).Usually, it is a conservative (worst-case oriented) methodology.
RO is useful if:Some parameters come from an estimation process and may be contaminated withestimation errors.There are “hard” constraints that must be satisfied no matter what.The objective function value/optimal solutions are highly sensitive to perturbations.The modeler/designer cannot afford low probability high-magnitude risks (typicalexample: designing a bridge).
M. Mera Trujillo Optimization Methods in Finance
![Page 30: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/30.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 31: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/31.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 32: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/32.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 33: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/33.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 34: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/34.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 35: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/35.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 36: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/36.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 37: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/37.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?
We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 38: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/38.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.
An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 39: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/39.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust Optimization
Main Contributors to Robust Optimization:
Kouvelis and Yu (minimax regret, Robust Discrete Optimization) [1].
Ben-Tal and Nemirovski (ellipsoidal uncertainty)[2].
Tutuncu and M. Koenig (Robust asset allocation) [3].
Major problems:
How to represent uncertainty?
How to compute a robust solution?
What is a robust solution anyway?We want to find a optimal (feasible) solution.An optimal solution is robust if it minimizes maximum relative regret.
M. Mera Trujillo Optimization Methods in Finance
![Page 40: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/40.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 41: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/41.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 42: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/42.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example:
Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 43: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/43.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph
where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 44: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/44.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty.
Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 45: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/45.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 46: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/46.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path:
Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 47: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/47.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds.
Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 48: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/48.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 49: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/49.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds),
that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 50: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/50.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Maximum relative regret
What is maximum relative regret?
Example: Consider the shortest path problem on a directed graph where arc costsare subject to uncertainty. Let us assume arc(i, j) can have as cost value anyvalue in the interval [cij , cij ].
Define the maximum relative regret associated with any path: Put all arc costs onthe path to their upper bounds and all other arc costs to their lower bounds. Findthe shortest path in this realization of arc costs.
Maximum Relative Regret = the cost of the path (at lower bounds), that of theshortest path.
M. Mera Trujillo Optimization Methods in Finance
![Page 51: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/51.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Uncertainty )
Data uncertainty or uncertainty in the parameters is describe through Uncertainty setsthat contain many possible values that may be realized for the uncertain parameters.The size of the uncertainty set is determined by the level of desired robustness.
M. Mera Trujillo Optimization Methods in Finance
![Page 52: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/52.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Uncertainty )
Data uncertainty or uncertainty in the parameters is describe through Uncertainty sets
that contain many possible values that may be realized for the uncertain parameters.The size of the uncertainty set is determined by the level of desired robustness.
M. Mera Trujillo Optimization Methods in Finance
![Page 53: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/53.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Uncertainty )
Data uncertainty or uncertainty in the parameters is describe through Uncertainty setsthat contain many possible values that may be realized for the uncertain parameters.
The size of the uncertainty set is determined by the level of desired robustness.
M. Mera Trujillo Optimization Methods in Finance
![Page 54: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/54.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition (Uncertainty )
Data uncertainty or uncertainty in the parameters is describe through Uncertainty setsthat contain many possible values that may be realized for the uncertain parameters.The size of the uncertainty set is determined by the level of desired robustness.
M. Mera Trujillo Optimization Methods in Finance
![Page 55: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/55.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 56: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/56.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 57: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/57.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters,
alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 58: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/58.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques,
among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 59: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/59.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 60: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/60.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 61: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/61.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 62: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/62.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 63: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/63.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Definition
Uncertainty sets can represent or may be formed by difference of opinions on futurevalues of certain parameters, alternative estimates of parameters generated viastatistical techniques from historical data and/or Bayesian techniques, among otherthings.
Types of uncertainty sets
Common types of uncertainty sets encountered in robust optimization models includethe following:
Uncertainty sets representing a finite number of scenarios generated for thepossible values of the parameters:
U = {p1, p2, ..., pk}
M. Mera Trujillo Optimization Methods in Finance
![Page 64: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/64.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing the convex hull of a finite number of scenarios
generated for the possible values of the parameters (these are sometimes calledpolytopic uncertainty sets):
U = conv(p1, p2, ..., pk )
M. Mera Trujillo Optimization Methods in Finance
![Page 65: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/65.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing the convex hull of a finite number of scenariosgenerated for the possible values of the parameters (these are sometimes calledpolytopic uncertainty sets):
U = conv(p1, p2, ..., pk )
M. Mera Trujillo Optimization Methods in Finance
![Page 66: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/66.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing the convex hull of a finite number of scenariosgenerated for the possible values of the parameters (these are sometimes calledpolytopic uncertainty sets):
U = conv(p1, p2, ..., pk )
M. Mera Trujillo Optimization Methods in Finance
![Page 67: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/67.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 68: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/68.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 69: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/69.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 70: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/70.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 71: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/71.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 72: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/72.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
Uncertainty sets representing an interval description for each uncertain parameter:
U = {p : l ≤ p ≤ u}
Confidence intervals encountered frequently in statistics can be the source ofsuch uncertainty sets.
Ellipsoidal uncertainty sets:
U = {p : p = p0 + M · u, ||u|| ≤ 1}
These uncertainty sets can also arise from statistical estimation in the form ofconfidence regions.
M. Mera Trujillo Optimization Methods in Finance
![Page 73: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/73.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline, the shape of the uncertainty set will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 74: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/74.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model
as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline, the shape of the uncertainty set will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 75: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/75.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline, the shape of the uncertainty set will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 76: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/76.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline,
the shape of the uncertainty set will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 77: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/77.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline, the shape of the uncertainty set
will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 78: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/78.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
To determine the uncertainty set that is appropriate for a particular model as wellas the type of uncertainty sets that lead to tractable problems, it is a non-trivialtask.
As a general guideline, the shape of the uncertainty set will often depend on thesources of uncertainty as well as the sensitivity of the solutions to theseuncertainties.
M. Mera Trujillo Optimization Methods in Finance
![Page 79: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/79.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 80: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/80.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.
For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 81: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/81.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example:
in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 82: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/82.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization,
uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 83: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/83.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables,
there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 84: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/84.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.
In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 85: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/85.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases,
after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 86: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/86.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes
we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 87: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/87.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 88: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/88.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Types of uncertainty sets
The size of the uncertainty set, on the other hand, will often be chosen based onthe desired level of robustness.For example: in mean-variance portfolio optimization, uncertain parameters reflectthe ”true” values of moments of random variables, there is no way of knowingthese unobservable true values exactly.In such cases, after making some assumptions about the stationarity of theserandom processes we can generate estimates of these true parameters usingstatistical procedures.
M. Mera Trujillo Optimization Methods in Finance
![Page 89: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/89.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 90: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/90.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets
and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 91: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/91.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression,
the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 92: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/92.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and
they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 93: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/93.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).
To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 94: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/94.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets,
use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 95: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/95.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 96: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/96.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 97: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/97.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set
can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 98: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/98.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated.
However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 99: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/99.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies,
their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 100: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/100.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Example:
Using a linear factor model for the multivariate returns of several assets and estimatethe factor loading matrices via linear regression, the confidence regions generated forthese parameters are ellipsoidal sets and they advocate their use in robust portfolioselection as uncertainty sets (Goldfarb and Iyengar).To generate interval type uncertainty sets, use bootstrapping strategies as well asmoving averages of returns from historical data (Tutuncu and Koenig).
NOTE:
The shape and the size of the uncertainty set can significantly affect the robustsolutions generated. However with few guidelines backed by theoretical and empiricalstudies, their choice remains an art form at the moment.
M. Mera Trujillo Optimization Methods in Finance
![Page 101: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/101.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 102: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/102.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 103: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/103.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints
and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 104: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/104.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.
For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 105: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/105.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example:
multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 106: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/106.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems
where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 107: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/107.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages
and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 108: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/108.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.
(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 109: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/109.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage)
nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 110: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/110.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.
The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 111: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/111.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Constraint Robustness
This refers to situations where the uncertainty is in the constraints and we seeksolutions that remain feasible for all possible values of the uncertain inputs.For example: multi-stage problems where the uncertain outcomes of earlier stageshave an effect on the decisions of the later stages and the decision variables must bechosen to satisfy certain balance constraints.(e.g., inputs to a particular stage can not exceed the outputs of the previous stage) nomatter what happens with the uncertain parameters of the problem.The solution must be constraint-robust with respect to the uncertainties of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 112: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/112.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions:
Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 113: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/113.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 114: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/114.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 115: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/115.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 116: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/116.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables,
f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 117: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/117.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function,
G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 118: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/118.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints
that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 119: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/119.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Mathematical model for finding constraint-robust solutions: Consider an optimizationproblem of the form:
minx
f (x)
G(x , p) ∈ K
x are the decision variables, f is the (certain) objective function, G and K are thestructural elements of the constraints that are assumed to be certain and p are thepossibly uncertain parameters of the problem.
M. Mera Trujillo Optimization Methods in Finance
![Page 120: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/120.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Consider an uncertainty set U
that contains all possible values of the uncertainparameters p.A constraint-robust optimal solution can be found by solving the following problem:
minx
f (x)
G(x , p) ∈ K ,∀p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 121: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/121.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Consider an uncertainty set U that contains all possible values of the uncertainparameters p.
A constraint-robust optimal solution can be found by solving the following problem:
minx
f (x)
G(x , p) ∈ K ,∀p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 122: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/122.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Consider an uncertainty set U that contains all possible values of the uncertainparameters p.A constraint-robust optimal solution can be found by solving the following problem:
minx
f (x)
G(x , p) ∈ K ,∀p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 123: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/123.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Consider an uncertainty set U that contains all possible values of the uncertainparameters p.A constraint-robust optimal solution can be found by solving the following problem:
minx
f (x)
G(x , p) ∈ K ,∀p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 124: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/124.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Mathematical model RO
Consider an uncertainty set U that contains all possible values of the uncertainparameters p.A constraint-robust optimal solution can be found by solving the following problem:
minx
f (x)
G(x , p) ∈ K ,∀p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 125: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/125.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust feasible set
The robust feasible set is the intersection of the feasible sets S(p) = {x : G(x , p) ∈ K}indexed by the uncertainty set U .
Figure: Constraint robustness
For an ellipsoidal feasible set with U = {p1, p2, p3, p4}, where pi correspond to theuncertain center of the ellipse.
M. Mera Trujillo Optimization Methods in Finance
![Page 126: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/126.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust feasible set
The robust feasible set is the intersection of the feasible sets S(p) = {x : G(x , p) ∈ K}indexed by the uncertainty set U .
Figure: Constraint robustness
For an ellipsoidal feasible set with U = {p1, p2, p3, p4}, where pi correspond to theuncertain center of the ellipse.
M. Mera Trujillo Optimization Methods in Finance
![Page 127: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/127.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust feasible set
The robust feasible set is the intersection of the feasible sets S(p) = {x : G(x , p) ∈ K}indexed by the uncertainty set U .
Figure: Constraint robustness
For an ellipsoidal feasible set with U = {p1, p2, p3, p4}, where pi correspond to theuncertain center of the ellipse.
M. Mera Trujillo Optimization Methods in Finance
![Page 128: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/128.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust feasible set
The robust feasible set is the intersection of the feasible sets S(p) = {x : G(x , p) ∈ K}indexed by the uncertainty set U .
Figure: Constraint robustness
For an ellipsoidal feasible set with U = {p1, p2, p3, p4},
where pi correspond to theuncertain center of the ellipse.
M. Mera Trujillo Optimization Methods in Finance
![Page 129: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/129.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Robust feasible set
The robust feasible set is the intersection of the feasible sets S(p) = {x : G(x , p) ∈ K}indexed by the uncertainty set U .
Figure: Constraint robustness
For an ellipsoidal feasible set with U = {p1, p2, p3, p4}, where pi correspond to theuncertain center of the ellipse.
M. Mera Trujillo Optimization Methods in Finance
![Page 130: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/130.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective Robustness
This refers to solutions that will remain close to optimal for all possible realizations ofthe uncertain problem parameters.
Since such solutions may be difficult to obtain, especially when uncertainty sets arerelatively large, an alternative goal for objective robustness is to find solutions whoseworst-case behavior is optimized.The worst-case behavior of a solution corresponds to the value of the objective functionfor the worst possible realization of the uncertain data for that particular solution.
M. Mera Trujillo Optimization Methods in Finance
![Page 131: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/131.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective Robustness
This refers to solutions that will remain close to optimal for all possible realizations ofthe uncertain problem parameters.Since such solutions may be difficult to obtain,
especially when uncertainty sets arerelatively large, an alternative goal for objective robustness is to find solutions whoseworst-case behavior is optimized.The worst-case behavior of a solution corresponds to the value of the objective functionfor the worst possible realization of the uncertain data for that particular solution.
M. Mera Trujillo Optimization Methods in Finance
![Page 132: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/132.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective Robustness
This refers to solutions that will remain close to optimal for all possible realizations ofthe uncertain problem parameters.Since such solutions may be difficult to obtain, especially when uncertainty sets arerelatively large,
an alternative goal for objective robustness is to find solutions whoseworst-case behavior is optimized.The worst-case behavior of a solution corresponds to the value of the objective functionfor the worst possible realization of the uncertain data for that particular solution.
M. Mera Trujillo Optimization Methods in Finance
![Page 133: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/133.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective Robustness
This refers to solutions that will remain close to optimal for all possible realizations ofthe uncertain problem parameters.Since such solutions may be difficult to obtain, especially when uncertainty sets arerelatively large, an alternative goal for objective robustness is to find solutions whoseworst-case behavior is optimized.
The worst-case behavior of a solution corresponds to the value of the objective functionfor the worst possible realization of the uncertain data for that particular solution.
M. Mera Trujillo Optimization Methods in Finance
![Page 134: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/134.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective Robustness
This refers to solutions that will remain close to optimal for all possible realizations ofthe uncertain problem parameters.Since such solutions may be difficult to obtain, especially when uncertainty sets arerelatively large, an alternative goal for objective robustness is to find solutions whoseworst-case behavior is optimized.The worst-case behavior of a solution corresponds to the value of the objective functionfor the worst possible realization of the uncertain data for that particular solution.
M. Mera Trujillo Optimization Methods in Finance
![Page 135: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/135.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness.
Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 136: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/136.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 137: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/137.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 138: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/138.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 139: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/139.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set
and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 140: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/140.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p.
As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 141: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/141.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p.
An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 142: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/142.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 143: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/143.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
To addresses objective robustness. Consider an optimization problem of the form:
minx
f (x , p)
x ∈ S
S is the (certain) feasible set and f is the objective function that depends on uncertainparameters p. As before, U denotes the uncertainty set that contains all possiblevalues of the uncertain parameters p. An objective robust solution can be obtained bysolving:
minx∈S
maxp∈U
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 144: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/144.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line,
the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 145: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/145.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5},
andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 146: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/146.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape.
Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 147: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/147.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure.
In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 148: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/148.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 149: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/149.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 150: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/150.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Objective robustness
The feasible set S is the real line, the uncertainty set is U = {p1, p2, p3, p4, p5}, andthe objective function f (x , pi ) is a convex quadratic function whose parameters pi
determine its shape. Note that the robust minimizer is different from the minimizers ofeach f (x , pi ) which are denoted by x∗i in the figure. In fact, none of the each x∗i isparticularly close to the robust minimizer.
Figure: Objective robustness
M. Mera Trujillo Optimization Methods in Finance
![Page 151: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/151.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 152: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/152.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 153: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/153.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 154: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/154.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 155: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/155.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 156: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/156.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables,
ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 157: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/157.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain),
f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 158: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/158.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions,
and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 159: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/159.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex
(e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 160: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/160.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What are we dealing with?
Consider the optimization problem:
min f (x , ξ)
subject to
gi (x , ξ) ≤ X
x is the vector of variables, ξ is the vector of data (uncertain), f and gi are convexfunctions, and X is a possibly non-convex (e.g., the set of nonnegative integers, orbinary integers) set.
M. Mera Trujillo Optimization Methods in Finance
![Page 161: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/161.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty, or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 162: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/162.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty, or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 163: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/163.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty, or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 164: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/164.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty, or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 165: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/165.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty,
or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 166: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/166.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming:
min cT · x
subject to
A · x ≥ b
A, b, and c could be plagued with uncertainty, or could be just estimates from asimulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 167: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/167.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 168: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/168.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 169: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/169.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 170: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/170.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 171: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/171.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 172: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/172.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers,
A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 173: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/173.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty,
or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 174: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/174.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Linear Programming with Integers:
min cT · x
subject to
A · x ≥ b
x ∈ X
X is the set of integers, or binary integers, A, b, and c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 175: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/175.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 176: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/176.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 177: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/177.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 178: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/178.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 179: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/179.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix.
Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 180: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/180.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty,
or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 181: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/181.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Our Typical Optimization Problems
Quadratic Programming:
min(
12
)· xT · Q · x + cT · x
subject to
A · x ≥ b
Q is a symmetric positive (semi-)definite matrix. Q, A, b, c could be plagued withuncertainty, or could be just estimates from a simulation or discretization process.
M. Mera Trujillo Optimization Methods in Finance
![Page 182: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/182.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 183: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/183.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 184: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/184.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 185: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/185.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 186: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/186.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 187: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/187.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 188: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/188.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 189: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/189.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 190: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/190.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Ben-Tal and Nemirovski Approach to Robust Optimization
Consider the linear program:
min cT · x
subject to
A · x ≥ b
Assume each row ai of A is uncertain but known to lie in ellipsoids.
Ei = {ai : ai = ai + Pi · ui , ||ui ||2 ≤ 1}
where Pi is symmetric positive (semi)definite matrix for all i .
Assume rows ai assume values independently of one another.
M. Mera Trujillo Optimization Methods in Finance
![Page 191: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/191.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What is a robust solution in the world of Ben-Tal and Nemirovski?
We want to make sure the constraints A · x ≥ b are satisfied for all realizations ofthe data.
We do not tolerate a violation of the constraints for any values of the uncertainparameters in the uncertainty set.
Among such solutions, pick one that minimizes the objective function value.
M. Mera Trujillo Optimization Methods in Finance
![Page 192: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/192.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What is a robust solution in the world of Ben-Tal and Nemirovski?
We want to make sure the constraints A · x ≥ b are satisfied for all realizations ofthe data.
We do not tolerate a violation of the constraints for any values of the uncertainparameters in the uncertainty set.
Among such solutions, pick one that minimizes the objective function value.
M. Mera Trujillo Optimization Methods in Finance
![Page 193: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/193.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What is a robust solution in the world of Ben-Tal and Nemirovski?
We want to make sure the constraints A · x ≥ b are satisfied for all realizations ofthe data.
We do not tolerate a violation of the constraints for any values of the uncertainparameters in the uncertainty set.
Among such solutions, pick one that minimizes the objective function value.
M. Mera Trujillo Optimization Methods in Finance
![Page 194: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/194.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
What is a robust solution in the world of Ben-Tal and Nemirovski?
We want to make sure the constraints A · x ≥ b are satisfied for all realizations ofthe data.
We do not tolerate a violation of the constraints for any values of the uncertainparameters in the uncertainty set.
Among such solutions, pick one that minimizes the objective function value.
M. Mera Trujillo Optimization Methods in Finance
![Page 195: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/195.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 196: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/196.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 197: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/197.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 198: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/198.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 199: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/199.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 200: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/200.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 201: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/201.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Consider a multi-period optimization problem with uncertain parameters whereuncertainty is revealed progressively through periods.
A subset of the decision variables can be chosen after observing realizations ofsome uncertain parameters.
This allows to correct the earlier decision made under a smaller information set.
In stochastic programming, this is called “recourse”
In robust optimization it is called “Adjustable Robust Optimization”(ARO).
Due to Ben-Tal, Guslitzer and Nemirovski.
M. Mera Trujillo Optimization Methods in Finance
![Page 202: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/202.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 203: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/203.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 204: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/204.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 205: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/205.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example:
Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 206: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/206.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now,
while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 207: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/207.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1,
A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 208: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/208.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 209: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/209.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
what is it?
Adjustable robust optimization (ARO) formulations model these decisionenvironment and allow recourse action.
ARO models were recently introduced for uncertain linear programming problems.
For example: Consider the two-stage linear optimization problem given belowwhose first-stage decision variables x1 need to be determined now, while thesecond-stage decision variables x2 can be chosen after the uncertain parametersof the problem A1, A2, and b are realized:
minx1,x2
{cT · x1 : A1 · x1 + A2 · x2 ≤ b}
M. Mera Trujillo Optimization Methods in Finance
![Page 210: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/210.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Assume that A1 is revealed to the modeler after choosing x1. So, at the momentof choosing x2 the modeler knows the value of A1.
Could we make robust choices of x2 taking this sequential nature of the decisionprocess into account?
M. Mera Trujillo Optimization Methods in Finance
![Page 211: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/211.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
ARO
Assume that A1 is revealed to the modeler after choosing x1. So, at the momentof choosing x2 the modeler knows the value of A1.
Could we make robust choices of x2 taking this sequential nature of the decisionprocess into account?
M. Mera Trujillo Optimization Methods in Finance
![Page 212: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/212.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 213: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/213.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2
that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 214: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/214.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 215: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/215.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 216: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/216.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 217: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/217.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 218: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/218.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 219: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/219.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness (standard robust counterpart)
Let U denote the uncertainty set for parameters A1, A2, and b.
The standard constraint robust optimization formulation for this problem seeks tofind vectors x1 and x2 that optimize the objective function and satisfy theconstraints of the problem for all possible realizations of the constraint coefficients.
Both sets of variables must be chosen before the uncertain parameters can beobserved and therefore cannot depend on these parameters.
The robust counterpart is (RC):
minx1{cT · x1 : ∃x2∀(A1,A2, b) ∈ U : A1 · x1 + A2 · x2 ≤ b}
Here notice that the choice of x2 is independent of the realized values of theuncertain parameters.
This ignores the multi-stage nature of the decision process.
M. Mera Trujillo Optimization Methods in Finance
![Page 220: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/220.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 221: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/221.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 222: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/222.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 223: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/223.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 224: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/224.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 225: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/225.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 226: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/226.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs.
Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 227: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/227.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robustness
The ARO formulation is (ARC):
minx1{cT · x1 : ∀(A1,A2, b) ∈ U ,∃x2 ≡ x2(A1,A2, b) : A1 · x1 + A2 · x2 ≤ b}
The feasible set of the second problem (ARO) is larger than the feasible set of theRC.
ARO is therefore less conservative compared to RC.
However, ARO is harder to formulate explicitly as an easy optimization probleme.g., linear programs.
Because we do not know the functional form of the dependency.
Only very simple uncertainty sets allow “nice” ARCs. Otherwise, we have toassume a simple functional form for dependencies, e.g., affine dependency.
M. Mera Trujillo Optimization Methods in Finance
![Page 228: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/228.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 229: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/229.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 230: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/230.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 231: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/231.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations,
c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 232: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/232.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients,
and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 233: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/233.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 234: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/234.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain,
i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 235: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/235.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub ,
where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 236: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/236.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
An Example from Network Design
Ordonez and Zhao considered the following network capacity expansion problemsubject to a budget constraint:
minx≥0,y≥0
{cT · x : N · x = b, x ≤ u + y , dT · ≤ I}
Where the vector y ∈ Rm denotes capacity expansion decisions, x ∈ Rm the flowvariables.
The constraints N · x = b represent the flow balance equations, c represents thetransportation cost coefficients, and d the cost of incremental unit capacity.
Assume c and b are uncertain, i.e., c ∈ Uc and b ∈ Ub , where Uc and Ub aresuitable (closed, bounded, convex) uncertainty sets.
M. Mera Trujillo Optimization Methods in Finance
![Page 237: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/237.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 238: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/238.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 239: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/239.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 240: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/240.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 241: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/241.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 242: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/242.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 243: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/243.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
IntroductionUncertainty setsRO ConceptsSummaryExampleAdjustable Robust Optimization (ARO)
Adjustable Robust Counterpart
The Adjustable Robust Counterpart is:
zARC = miny,γ
γ
subject to
dT · x ≤ I, y ≥ 0,
∀c ∈ Uc , b ∈ Ub ∃x :
N · x = b0 ≤ x ≤ u + yCT · x ≤ γ
Ordonez and Zhao proved :
zARC = miny≥0,dT ·y≤I
maxc∈Uc ,b∈Ub
minN·x=b,0≤x≤u+y
M. Mera Trujillo Optimization Methods in Finance
![Page 244: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/244.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 245: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/245.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 246: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/246.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty
is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 247: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/247.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 248: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/248.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters
and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 249: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/249.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 250: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/250.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints,
we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 251: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/251.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 252: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/252.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
One of the simplest strategies for achieving robustness under uncertainty is tosample several scenarios for the uncertain parameters from a set that containspossible values of these parameters.
Sampling can be done with or without using distributional assumptions on theparameters and produces a robust optimization formulation with a finiteuncertainty set.
If uncertain parameters appear in the constraints, we create a copy of each suchconstraint corresponding to each scenario.
Uncertainty in the objective function can be handled in a similar manner.
M. Mera Trujillo Optimization Methods in Finance
![Page 253: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/253.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Consider the generic uncertain optimization problem:
minx
f (x)
G(x , p) ∈ K
M. Mera Trujillo Optimization Methods in Finance
![Page 254: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/254.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Consider the generic uncertain optimization problem:
minx
f (x)
G(x , p) ∈ K
M. Mera Trujillo Optimization Methods in Finance
![Page 255: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/255.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Consider the generic uncertain optimization problem:
minx
f (x)
G(x , p) ∈ K
M. Mera Trujillo Optimization Methods in Finance
![Page 256: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/256.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Consider the generic uncertain optimization problem:
minx
f (x)
G(x , p) ∈ K
M. Mera Trujillo Optimization Methods in Finance
![Page 257: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/257.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
If the uncertainty set U is a a finite set, i.e., U = {p1, p2, ..., pk} the robustformulation is obtained as follows:
mint,x
t
t − f (x , pi ) ≥ 0, i = 1, ..., k ,
G(x , p1) ∈ K , i = 1, ..., k
M. Mera Trujillo Optimization Methods in Finance
![Page 258: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/258.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
If the uncertainty set U is a a finite set, i.e., U = {p1, p2, ..., pk} the robustformulation is obtained as follows:
mint,x
t
t − f (x , pi ) ≥ 0, i = 1, ..., k ,
G(x , p1) ∈ K , i = 1, ..., k
M. Mera Trujillo Optimization Methods in Finance
![Page 259: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/259.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
If the uncertainty set U is a a finite set, i.e., U = {p1, p2, ..., pk} the robustformulation is obtained as follows:
mint,x
t
t − f (x , pi ) ≥ 0, i = 1, ..., k ,
G(x , p1) ∈ K , i = 1, ..., k
M. Mera Trujillo Optimization Methods in Finance
![Page 260: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/260.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
If the uncertainty set U is a a finite set, i.e., U = {p1, p2, ..., pk} the robustformulation is obtained as follows:
mint,x
t
t − f (x , pi ) ≥ 0, i = 1, ..., k ,
G(x , p1) ∈ K , i = 1, ..., k
M. Mera Trujillo Optimization Methods in Finance
![Page 261: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/261.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
If the uncertainty set U is a a finite set, i.e., U = {p1, p2, ..., pk} the robustformulation is obtained as follows:
mint,x
t
t − f (x , pi ) ≥ 0, i = 1, ..., k ,
G(x , p1) ∈ K , i = 1, ..., k
M. Mera Trujillo Optimization Methods in Finance
![Page 262: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/262.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Note that no reformulation is necessary in this case and the duplicated constraintspreserve the structural properties (linearity, convexity, etc.) of the originalconstraints.
M. Mera Trujillo Optimization Methods in Finance
![Page 263: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/263.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Note that no reformulation is necessary in this case
and the duplicated constraintspreserve the structural properties (linearity, convexity, etc.) of the originalconstraints.
M. Mera Trujillo Optimization Methods in Finance
![Page 264: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/264.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Sampling
Note that no reformulation is necessary in this case and the duplicated constraintspreserve the structural properties (linearity, convexity, etc.) of the originalconstraints.
M. Mera Trujillo Optimization Methods in Finance
![Page 265: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/265.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Outline
1 Theory of Robust Optimization
Introduction
Uncertainty sets
RO Concepts
Summary
Example
Adjustable Robust Optimization
(ARO)2 Tools and Strategies for Robust
Optimization
Sampling
Saddle-Point Characterizations
3 Bibliography
M. Mera Trujillo Optimization Methods in Finance
![Page 266: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/266.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 267: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/267.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions
when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 268: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/268.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 269: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/269.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms
such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 270: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/270.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 271: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/271.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:
minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 272: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/272.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 273: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/273.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem
is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 274: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/274.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 275: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/275.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
The robust solution can be characterized using saddle-point conditions when theoriginal problem satisfies certain convexity assumptions.
The benefit of this characterization is that we can then use algorithms such asinterior-point methods already developed and available for saddle-point problems.
Consider:minx∈S
maxp∈U
f (x , p)
Note that the dual of this robust optimization problem is obtained by changing theorder of the minimization and maximization problems:
maxp∈U
minx∈S
f (x , p)
M. Mera Trujillo Optimization Methods in Finance
![Page 276: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/276.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p, if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 277: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/277.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p, if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 278: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/278.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p,
if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 279: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/279.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p, if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and
there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 280: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/280.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p, if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 281: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/281.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Saddle-Point Characterizations
From standard results in convex analysis we have the following conclusion:
Lemma: If f (x , p) is a convex function of x and concave function of p, if S and U arenonempty and at least one of them is bounded the optimal values of the problemsabove and there exists a saddle point (x∗, p∗) such that:
f (x∗, p) ≤ f (x∗, p∗) ≤ f (x , p∗), ∀x ∈ S, p ∈ U
M. Mera Trujillo Optimization Methods in Finance
![Page 282: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/282.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Software
Robust Optimization and Robust Programming software:
http://www.aimms.com/operations-research/
mathematical-programming/robust-optimization/
Conclusion
Robustness = best solution against the worst possible data realization.
M. Mera Trujillo Optimization Methods in Finance
![Page 283: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/283.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Software
Robust Optimization and Robust Programming software:http://www.aimms.com/operations-research/
mathematical-programming/robust-optimization/
Conclusion
Robustness = best solution against the worst possible data realization.
M. Mera Trujillo Optimization Methods in Finance
![Page 284: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/284.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
SamplingSaddle-Point Characterizations
Software
Robust Optimization and Robust Programming software:http://www.aimms.com/operations-research/
mathematical-programming/robust-optimization/
Conclusion
Robustness = best solution against the worst possible data realization.
M. Mera Trujillo Optimization Methods in Finance
![Page 285: Robust Optimization: Theory and Toolscommunity.wvu.edu/.../sp15/optfin/lecnotes/RobustO_Theo.pdfOutline Robust Optimization: Theory and Tools Marcela Mera Trujillo1 1Math Department](https://reader033.vdocument.in/reader033/viewer/2022042021/5e78449ac977e758c32280da/html5/thumbnails/285.jpg)
Theory of Robust OptimizationTools and Strategies for Robust Optimization
Bibliography
P. Kouvelis and G. Yu. Robust Discrete Optimization and its Applications. KluwerAcademic Publishers, Amsterdam, 1997.
A. Ben-Tal, T. Margalit, and A. N. Nemirovski. Robust modeling of multi-stageportfolio problems. In H. Frenk, K. Roos, T. Terlaky, and S. Zhang, editors, HighPerformance Optimization, pages 303-328.Kluwer Academic Publishers, 2002.
R. H. Tutuncu and M. Koenig. Robust asset allocation. Annals of OperationsResearch, 132:157-187, 2004.
Mustafa Pinar. Course taught, on-line:http://www.ie.bilkent.edu.tr/˜mustafap/courses/
G. Cornuejols and R. Tutuncu. Optimization Methods in Finance. Carnegie MellonUniversity, Pittsburgh, 2006.
M. Mera Trujillo Optimization Methods in Finance