bv_cvxbook_extra_exercises

n where we minimize over all possible probability

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Unformatted text preview: Ps (P) minimize cT x subject to F (x) (D) maximize tr(F0 Z ) subject to tr(Fi Z ) + ci = 0, Z 0, 0 i = 1, . . . , m where F (x) = F0 + x1 F1 + · · · + xn Fn and Fi ∈ Sp for i = 0, . . . , n. Let Z ⋆ be a solution of (D). Show that every solution x⋆ of the unconstrained problem minimize cT x + M max{0, λmax (F (x))}, where M > tr Z ⋆ , is a solution of (P). 4.9 Quadratic penalty. Consider the problem minimize f0 (x) subject to fi (x) ≤ 0, i = 1, . . . , m, (7) where the functions fi : Rn → R are differentiable and convex. Show that m φ ( x ) = f0 ( x ) + α i=1 max{0, fi (x)}2 , where α > 0, is convex. Suppose x minimizes φ. Show how to find from x a feasible point for the ˜ ˜ dual of (7). Find the corresponding lower bound on the optimal value of (7). 4.10 Binary least-squares. We consider the non-convex least-squares approximation problem with binary constraints minimize Ax − b 2 2 (8) subject to x2 = 1, k = 1, . . . , n, k 30 where A ∈ Rm×n and b ∈ Rm . We ass...
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