v2010.10.26 - Convex Optimization

246 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSconfined to the entire positive semidefinite cone (including its boundary). Itis now our goal to incorporate f into an optimization problem such thatan optimal solution returned always comprises a projection matrix W . Theset of orthogonal projection matrices is a nonconvex subset of the positivesemidefinite cone. So f cannot be convex on the projection matrices, andits equivalent (for idempotent W )f(W , x) = x T( I − (1 + ǫ) −1 W ) x (572)cannot be convex simultaneously in both variables on either the positivesemidefinite or symmetric projection matrices.Suppose we allow domf to constitute the entire positive semidefinitecone but constrain W to a Fantope (90); e.g., for convex set C and 0 < k < nas inminimize ǫx T (W + ǫI) −1 xx∈R n , W ∈S nsubject to 0 ≼ W ≼ I(573)trW = kx ∈ CAlthough this is a convex problem, there is no guarantee that optimal W isa projection matrix because only extreme points of a Fantope are orthogonalprojection matrices UU T .Let’s try partitioning the problem into two convex parts (one for x andone for W), substitute equivalence (570), and then iterate solution of convexproblemminimize x T (I − (1 + ǫ) −1 W)xx∈R n(574)subject to x ∈ Cwith convex problem(a)minimize x ⋆T (I − (1 + ǫ) −1 W)x ⋆W ∈S nsubject to 0 ≼ W ≼ ItrW = k≡maximize x ⋆T Wx ⋆W ∈S nsubject to 0 ≼ W ≼ ItrW = k(575)until convergence, where x ⋆ represents an optimal solution of (574) fromany particular iteration. The idea is to optimally solve for the partitionedvariables which are later combined to solve the original problem (573).What makes this approach sound is that the constraints are separable, the