bv_cvxbook_extra_exercises

In this problem you will use a typical method to

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Unformatted text preview: and x ∈ Rn is the variable. For simplicity, we will assume that dom fi = Rn for each i. Now introduce a new scalar variable t ∈ R and form the convex problem minimize cT x subject to tfi (x/t) ≤ 0, i = 1, . . . , m, Ax = b, t = 1. Define K = cl{(x, t) ∈ Rn+1 | tfi (x/t) ≤ 0, i = 1, . . . , m, t > 0}. Then our original problem can be expressed as minimize cT x subject to (x, t) ∈ K, Ax = b, t = 1. This is a conic problem when K is proper. You will relate some properties of the original problem to K . (a) Show that K is a convex cone. (It is closed by definition, since we take the closure.) (b) Suppose the original problem is strictly feasible, i.e., there exists a point x with fi (x) < 0, ¯ i = 1, . . . , m. (This is called Slater’s condition.) Show that K has nonempty interior. (c) Suppose that the inequalities define a bounded set, i.e., {x | fi (x) ≤ 0, i = 1, . . . , m} is bounded. Show that K is pointed. 3.16 Exploring nearly optimal points. An optimization algorithm will find an optimal point for a problem,...
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