CSE291 Convex Optimization: Problem Statement CK Cheng Dept. of - - PowerPoint PPT Presentation

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CSE291 Convex Optimization: Problem Statement

CK Cheng

• Dept. of Computer Science and Engineering

University of California, San Diego

Outlines

• General Convex Problem Formats
• Convex Sets

– Specification – Sets and Definitions

• Convex Functions
• Convex Optimization Problems

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General Formats

min f0(x) subject to fi(x)≤bi, i=1, …, m, where functions f0,…,fm: Rn→R are convex, i.e.

• fi(αx+βy) ≤αfi(x)+βfi(y)
• for all x, y ϵ Rn and all α, β ϵ R with α+β=1, α≥0,

β≥0.

Examples:

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General Formats

min f0(x) subject to fi(x)≤bi, i=1, …, m,

• f0 is a convex function
• {x| fi(x) ≤bi} is a convex set for all i=1, …,m

Convex set C: for all x, yϵ C

• αx+βy ϵ C, for all α+β=1, and α, β≥0.

Examples:

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Outlines

• General Convex Problem Formats
• Convex Sets

– Specification

Sets and Definitions

• Affine Sets, Cones, Convex Hulls
• Hyperplanes and Half Spaces
• Polyhedra
• Matrix Positive Semidefinite Cones
• Dual Cones

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Convex Set Specification

We can describe the convex sets using

• Implicit Expression (equations)
• Explicit Expression (enumerations)

Examples:

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Explicit Expression: Examples

{θ1u1+θ2u2+…+θkuk | θ1+θ2+…+θk=1, θi≥0, for all i}

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Sets and Definitions

• Affine Sets, Cones, Convex Hulls
• Hyperplanes and Half Spaces
• Polyhedra (poly + hedron)
• Matrix Positive Semidefinite Cones
• Dual Cones

Examples

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