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

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

3. 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:

4. 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:

5. 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

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

7. Explicit Expression: Examples
{θ1u1+θ2u2+…+θkuk | θ1+θ2+…+θk=1, θi≥0, for all i}

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