quantile regression lecture 1 - pku · brewery problem observation. regardless of objective...
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![Page 1: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/1.jpg)
Quantile Regression
Ruibin Xi
![Page 2: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/2.jpg)
Computation of Quantile Regression (1)
• We have
• Minimizing is equivalent to minimizing
with constraints
![Page 3: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/3.jpg)
Computation of Quantile Regression (2)
• If we write
• the above problem can be written as
• Note that this is a linear programming (LP) problem
![Page 4: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/4.jpg)
Linear Programming (1)
• Standard maximum problem
– Optimize a linear function subject to linear inequalities
(P) Maximize
Subject to
![Page 5: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/5.jpg)
Linear Programming (2)
• Standard minimum problem
– Optimize a linear function subject to linear inequalities
(P) Minimize
Subject to
![Page 6: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/6.jpg)
Brewery Example (1)
![Page 7: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/7.jpg)
Brewery Example (2)
![Page 8: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/8.jpg)
Standard LP forms
• Standard maximum (equality) form
• Standard minimum (equality) form
![Page 9: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/9.jpg)
Brewery Example: converting to standard LP form
![Page 10: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/10.jpg)
Equivalent form
![Page 11: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/11.jpg)
Brewery problem: feasible region
![Page 12: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/12.jpg)
Brewery problem: Objective function
![Page 13: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/13.jpg)
Brewery problem: Geometry
![Page 14: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/14.jpg)
Convexity
![Page 15: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/15.jpg)
Vertices (1)
![Page 16: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/16.jpg)
Vertices (2)
![Page 17: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/17.jpg)
Basic feasible solutions(1)
![Page 18: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/18.jpg)
Basic feasible solutions(2)
![Page 19: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/19.jpg)
Basic feasible solutions(3)
![Page 20: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/20.jpg)
Basic feasible solutions(4)
![Page 21: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/21.jpg)
Basic feasible solutions(5)
![Page 22: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/22.jpg)
Basic feasible solutions(6)
![Page 23: Quantile Regression Lecture 1 - PKU · Brewery problem observation. Regardless of objective function coefficients, an optimal solution occurs at a vertex (0, 32) Beer (1 2, 28) (26,](https://reader036.vdocument.in/reader036/viewer/2022062604/5fc19f5a9fd8e231e240d735/html5/thumbnails/23.jpg)
Simplex algorithm