v2007.09.13 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 2173.2.2.0.1 Example. Matrix fractional function redux.Generalizing Example 3.1.7.0.4 consider a matrix-valued function of twovariables on domg = S N + ×R n×N for small positive constant ǫ (confer (1634))g(A, X) = ǫX(A + ǫI) −1 X T (530)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×N : Consider Schur form (1310) fromA.4: for T ∈ S n[ A + ǫI XTG(A, X, T ) =X ǫ −1 T⇔T − ǫX(A + ǫI) −1 X T ≽ 0A + ǫI ≻ 0]≽ 0(531)By Theorem 2.1.9.0.1, inverse image of the positive semidefinite cone S N+n+under affine mapping G(A, X, T ) is convex. Function g(A, X) is convexon S N + ×R n×N because its epigraph is that inverse image:epig(A, X) = { (A, X, T ) | A + ǫI ≻ 0, ǫX(A + ǫI) −1 X T ≼ T } = G −1( )S N+n+(532)3.2.3 second-order condition, matrix functionThe following line theorem is a potent tool for establishing convexity of amultidimensional function. To understand it, what is meant by line must firstbe solidified. Given a function g(X) : R p×k →S M and particular X, Y ∈ R p×knot necessarily in that function’s domain, then we say a line {X+ t Y | t ∈ R}passes through domg when X+ t Y ∈ domg over some interval of t ∈ R .3.2.3.0.1 Theorem. Line theorem. [46,3.1.1]Matrix-valued function g(X) : R p×k →S M is convex in X if and only if itremains convex on the intersection of any line with its domain. ⋄Now we assume a twice differentiable function.