v2010.10.26 - Convex Optimization

3.7. CONVEX MATRIX-VALUED FUNCTION 265For each and every X,Y ∈ domg (confer (614))g(Y ) ≽g(X) +→Y −Xdg(X) (628)S M +must therefore be necessary and sufficient for convexity of a matrix-valuedfunction of matrix variable on open convex domain.3.7.2 epigraph of matrix-valued function, sublevel setsWe generalize epigraph to a continuous matrix-valued function [34, p.155]g(X) : R p×k →S M :epi g {(X , T )∈ R p×k × S M | X ∈ domg , g(X) ≼T } (629)S M +from which it followsg convex ⇔ epig convex (630)Proof of necessity is similar to that in3.5 on page 242.Sublevel sets of a convex matrix-valued function corresponding to eachand every S ∈ S M (confer (559))L Sg {X ∈ domg | g(X) ≼S } ⊆ R p×k (631)S M +are convex. There is no converse.3.7.2.0.1 Example. Matrix fractional function redux. [34, p.155]Generalizing Example 3.5.0.0.4 consider a matrix-valued function of twovariables on domg = S N + ×R n×N for small positive constant ǫ (confer (1870))g(A, X) = ǫX(A + ǫI) −1 X T (632)where the inverse always exists by (1451). This function is convexsimultaneously in both variables over the entire positive semidefinite cone S N +