v2010.10.26 - Convex Optimization

3.5. EPIGRAPH, SUBLEVEL SET 245where the inverse always exists by (1451). This function is convexsimultaneously in both variables over the entire positive semidefinite cone S n +and all x∈ R n : Consider Schur-form (1514) fromA.4: for t ∈ R[ ] A + ǫI zG(A, z , t) =z T ǫ −1 ≽ 0t⇔t − ǫz T (A + ǫI) −1 z ≥ 0A + ǫI ≻ 0(567)Inverse image of the positive semidefinite cone S n+1+ under affine mappingG(A, z , t) is convex by Theorem 2.1.9.0.1. Function f(A, z) is convex onS+×R n n because its epigraph is that inverse image:epi f(A, z) = { (A, z , t) | A + ǫI ≻ 0, ǫz T (A + ǫI) −1 z ≤ t } = G −1( )S n+1+(568)3.5.1 matrix fractional projector functionConsider nonlinear function f having orthogonal projector W as argument:f(W , x) = ǫx T (W + ǫI) −1 x (569)Projection matrix W has property W † = W T = W ≽ 0 (1916). 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 (1870) 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 (570)limǫ→0 +f(W, x) = lim (W + ǫI) −1 x = x T (I − W )x (571)ǫ→0 +ǫxTwhere I − W is also an orthogonal projector (E.2).We learned from Example 3.5.0.0.4 that f(W , x)= ǫx T (W +ǫI) −1 x isconvex simultaneously in both variables over all x ∈ R n when W ∈ S n + is