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

CSE291 Convex Optimization: Problem Statement CK Cheng Dept. of Computer Science and Engineering University of California, San Diego 1

Outlines • General Convex Problem Formats • Convex Sets – Specification – Sets and Definitions • Convex Functions • Convex Optimization Problems 2

General Formats min f 0 (x) subject to f i (x)≤b i , i=1, …, m, where functions f 0 ,…,f m : R n →R are convex, i.e. • f i ( α x+ βy) ≤α f i (x)+ β f i (y) • for all x, y ϵ R n and all α , β ϵ R with α + β =1, α≥ 0, β≥ 0. Examples: 3

General Formats min f 0 (x) subject to f i (x)≤b i , i=1, …, m, • f 0 is a convex function • {x| f i (x) ≤b i } is a convex set for all i=1, …,m Convex set C: for all x, y ϵ C • α x+ β y ϵ C, for all α + β =1, and α , β≥ 0. Examples: 4

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 5

Convex Set Specification We can describe the convex sets using • Implicit Expression (equations) • Explicit Expression (enumerations) Examples: 6

Explicit Expression: Examples { θ 1 u 1 + θ 2 u 2 +…+ θ k u k | θ 1 + θ 2 +…+ θ k =1, θ i ≥ 0, for all i} 7

Sets and Definitions • Affine Sets, Cones, Convex Hulls • Hyperplanes and Half Spaces • Polyhedra (poly + hedron) • Matrix Positive Semidefinite Cones • Dual Cones Examples 8

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