v2007.09.13 - Convex Optimization

3.2. MATRIX-VALUED CONVEX FUNCTION 2153.2.0.0.1 Definition. Convex matrix-valued function:1) Matrix-definition.A function g(X) : R p×k →S M is convex in X iff domg is a convex set and,for each and every Y,Z ∈domg and all 0≤µ≤1 [157,2.3.7]g(µY + (1 − µ)Z) ≼µg(Y ) + (1 − µ)g(Z) (521)S M +Reversing the sense of the inequality flips this definition to concavity. Strictconvexity is defined less a stroke of the pen in (521) similarly to (422).2) Scalar-definition.It follows that g(X) : R p×k →S M is convex in X iff w T g(X)w : R p×k →R isconvex in X for each and every ‖w‖= 1; shown by substituting the defininginequality (521). By dual generalized inequalities we have the equivalent butmore broad criterion, (2.13.5)g convex ⇔ 〈W , g〉 convex∀W ≽S M +0 (522)Strict convexity on both sides requires caveat W ≠ 0. Because the set ofall extreme directions for the positive semidefinite cone (2.9.2.4) comprisesa minimal set of generators for that cone, discretization (2.13.4.2.1) allowsreplacement of matrix W with symmetric dyad ww T as proposed. △3.2.1 first-order convexity condition, matrix functionFrom the scalar-definition we have, for differentiable matrix-valuedfunction g and for each and every real vector w of unit norm ‖w‖= 1,w T g(Y )w ≥ w T →Y −XTg(X)w + w dg(X) w (523)that follows immediately from the first-order condition (509) for convexity ofa real function because→Y −XTw dg(X) w = 〈 ∇ X w T g(X)w , Y − X 〉 (524)respect to the nonnegative orthant R m×k+ . Symmetric convex functions share the samebenefits as symmetric matrices. Horn & Johnson [149,7.7] liken symmetric matrices toreal numbers, and (symmetric) positive definite matrices to positive real numbers.