v2007.09.13 - Convex Optimization

198 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSwhose epigraph isepif(A, x) = { (A, x, t) | A ≽ 0, x T Ax ≤ t } (468)Provide two simple explanations why f(A, x) = x T Ax is not a functionconvex simultaneously in positive semidefinite matrix A and vector x ondomf = S+× n R n .3.1.7.0.4 Example. Matrix fractional function. (confer3.2.2.0.1)Continuing Example 3.1.7.0.2, now consider a real function of two variableson domf = S n +×R n for small positive constant ǫ (confer (1634))f(A, x) = ǫx T (A + ǫI) −1 x (469)where the inverse always exists by (1251). This function is convexsimultaneously in both variables over the entire positive semidefinite cone S n +and all x∈ R n : Consider Schur form (1310) 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(470)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:epif(A, z) = { (A, z , t) | A + ǫI ≻ 0, ǫz T (A + ǫI) −1 z ≤ t } = G −1( S n+1+(471)3.1.7.1 matrix fractional projector functionConsider nonlinear function f having orthogonal projector W as argument:f(W , x) = ǫx T (W + ǫI) −1 x (472)Projection matrix W has property W † = W T = W ≽ 0 (1678). Anyorthogonal projector can be decomposed into an outer product of)