Convex optimization midterm

Convex Optimization. Midterm Exam. Question 1(Finite Cover of a Convex Set). Suppose a compact convex set C ∈ Rd is covered by a family F of open half-spaces 
Show how to computer the certifcate λ, if it exists, via linear programming. Solution ( Method 1 ). Let's prove this in two steps. 2Recall that Radon's Theorem.

Can a convex function have more than one optimal point?

In other word, the convex function has to have only one optimal value, but the optimal point does not have to be one

The below loosely convex function has one optimal value with multiple optimal points

What is a convex optimization problem?

A convex optimization problem is one in which the objective and constraintfunctionsareconvex,whichmeanstheysatisfytheinequality fi(fix+fly)•fifi(x)+flfi(y) (1

3) 2 1 Introduction forallx;y2Rnandallfi;fl2Rwithfi+fl=1,fi‚0,fl‚0


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