ee563 convex optimization - module 2 · second-order cone programming zubair khalid department of...
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Second-Order Cone Programming
Zubair Khalid
Department of Electrical EngineeringSchool of Science and Engineering
Lahore University of Management Sciences
https://www.zubairkhalid.org/ee563_2020.html
Convex Optimization
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Outline
• SOCP Formulation
• QCQP as SOCP
• Example: Robust Linear Program
Section 4.4.2
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Second-Order Cone Programming (SOCP)
Formulation:
Second-order Cone Constraint:
Here:
Let’s look at the inequality constraint:
This can be interpreted as if the vector
lies in a second-order (Quadratic, Lorentz) cone.
Recall:
In some texts, the constraint may also be expressed as:
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Second-Order Cone Programming (SOCP)QCQP – a special case of SOCP
SOCP: QCQP:
Let’s look at the inequality constraint:Using epigraph reformulation:
When c=0, the constraint becomes
This is a quadratic constraint; squaring it converts it into Quadratic constraint. SOCP becomes QCQP.
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Second-Order Cone Programming (SOCP)QCQP – a special case of SOCP
In QCQP reformulation, we have constraints:
Q: Are they equivalent?
Q: How can we represent Quadratic constraint as second-order constraint?
Or equivalently can berepresented as second-order constraint
So, they are equivalent
By defining
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Example: Robust Linear ProgramLinear Program:
Problem Under Consideration: Uncertainty in the constraints
(Already studied)
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Example: Robust Linear Program
Q: How can we incorporate this probabilistic constraint in the formulation of optimization problem?
A: This can be formulated as a second-order constraint.
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Example: Robust Linear Program
Assume
Also define
A Not-for-Profit University
--r q. ?( l
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Example: Robust Linear Program
Probabilistic constraint as second-order constraint
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Feedback: Questions or Comments?
Email: [email protected]
Slides available at: https://www.zubairkhalid.org/ee563_2020.html(Let me know should you need latex source)