Conic Optimization and Semidefinite 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

Outline

• Convex Optimization with Generalized Inequality Constraints

• Conic Optimization Problems

• Semidefinite Programming (SDP) Section 4.6

Convexity w.r.t. Generalized Inequality

Generalized Inequality:

Convexity w.r.t. Generalized Inequality:

Convex Optimization with Generalized Inequality Constraints:

Ordinary convex optimization problem

- If we allow the constraint functions to be vector valued:

Convex Optimization with Generalized Inequality Constraints

Convex Optimization with Generalized Inequality Constraints:

Properties and Interpretation

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Conic Form Problems

In convex Optimization with Generalized Inequality Constraints, if we consider

• Objective function: Affine (or linear) • Inequality constraints: Affine

Conic Form Problem or Conic Optimization Problem or Conic Program (CP)

- Conic form problems; generalization of LP

- The cones which we have a tractable conic optimization are of three basic types.

Conic Form Problems

- Component-wise inequality- Linear inequality constraints - half-spaces- Conic optimization problem becomes a linear program

- We have SOCP

Formulation of second-order constraint as conic constraint:

- Conic optimization problem is referred to as Semidefinite Program (SDP)

Semidefinite Program (SDP)

Formulation:

Notes:

Block Diagonal Matrix

Semidefinite Program (SDP)

m1n1m1ze T ~

i == 1, 2, ... ,m

• If matrices defining any LMI, the LMI reduces to a linear inequality con­straint (half-space).

• Multiple LMI constraints can be formulated as a single LMI.

i == 1, 2, ... ,m ---+

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SOCP as SDP:

Semidefinite Program (SDP)

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X:::.. A ..B s•

c_ -

f>-o :ltlf A ;..o,

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)( ~0<9 s ~0

Semidefinite Program (SDP)

SOC as LMI:

SOCP as SDP:

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(A~_,_byr e ~~..,.J. -t. i ~I - .. -w'll x'

?;o --- -~-~1-~--------- ----- ?;o A1... -+b

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s ::=. t.:r~ -

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Conic Form Problem or Conic Program (CP)

Convex Optimization with Generalized Inequality Constraints

• LP• SOCP• QCQP• QP• LP

• SDP• SOCP• QCQP• QP• LP

Big Picture: Connecting the Dots

Generalization of LP

Feedback: Questions or Comments?

Email: zubair.khalid@lums.edu.pk

Slides available at: https://www.zubairkhalid.org/ee563_2020.html(Let me know should you need latex source)