v2007.11.26 - Convex Optimization

200 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.1.7.1 matrix fractional projector functionConsider nonlinear function f having orthogonal projector W as argument:f(W , x) = ǫx T (W + ǫI) −1 x (473)Projection matrix W has property W † = W T = W ≽ 0 (1709). Anyorthogonal projector can be decomposed into an outer product oforthonormal matrices W = UU T where U T U = I as explained inE.3.2. From (1665) for any ǫ > 0 and idempotent symmetric W ,ǫ(W + ǫI) −1 = I − (1 + ǫ) −1 W from whichThereforef(W , x) = ǫx T (W + ǫI) −1 x = x T( I − (1 + ǫ) −1 W ) x (474)limǫ→0 +f(W, x) = lim (W + ǫI) −1 x = x T (I − W )x (475)ǫ→0 +ǫxTwhere I − W is also an orthogonal projector (E.2).We learned from Example 3.1.7.0.4 that f(W , x)= ǫx T (W +ǫI) −1 x isconvex simultaneously in both variables over all x ∈ R n when W ∈ S n + isconfined 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 (476)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 (80); 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(477)trW = kx ∈ C