Ho Chi Minh City University of Technology

Faculty of Electrical and Electronics Engineering

Department of Telecommunications

Lectured by Ha Hoang Kha, Ph.D.

Ho Chi Minh City University of Technology

Email: hahoangkha@gmail.com

Convex Optimization

Content

Convex optimization 2 H. H. Kha

References

Slides are adapted from the following resources:

D. P. Palomar, Convex Optimization, lecture notes.

Stephen Boyd and Lieven Vandenberghe, Convex

Optimization, Cambridge, U.K.: Cambridge University

Press, 2004.

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Convex Set

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Definition of Convex Set

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Examples: Hyperplane and halfspace

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Example: Polyhedra

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Examples: Euclidean Balls and Ellipsoids

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Convex Combination and Convex Hull

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Convex Cones

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Norm Balls and Norm Cones

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Positive Semidenite Cone

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Operations that Preserve Convexity

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Intersection

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Affine Function

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Convex Functions

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Definition of Convex Functions

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Example on R

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Example on Rn

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Example on Rnxn

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Epigraph

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Restriction of a Convex Function to a Line

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Example

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First and Second Order Condition

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First and Second Order Condition

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Example

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Example

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Operations that Preserve Convexity

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Operations that Preserve Convexity

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Operations that Preserve Convexity

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Operations that Preserve Convexity

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Operations that Preserve Convexity

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Convex Optimization

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Optimization Problem in Standard Form

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Optimization Problem in Standard Form

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Global and Local Optimality

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Implicit Constraints

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Feasibility Problem

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Convex Optimization Problem

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Example

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Global and Local Optimality

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Equivalent Reformulations

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Equivalent Reformulations

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Equivalent Reformulations

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Linear Programming (LP)

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Norm Problems

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Norm Problems

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Quadratic Programming (QP)

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Quadratically Constrained QP (QCQP)

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Second-Order Cone Programming (SOCP)

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Semidenite Programming (SDP)

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Semidenite Programming (SDP)

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