v2007.11.26 - Convex Optimization

216 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.2 Matrix-valued convex functionWe need different tools for matrix argument: We are primarily interested incontinuous matrix-valued functions g(X). We choose symmetric g(X)∈ S Mbecause matrix-valued functions are most often compared (522) with respectto the positive semidefinite cone S M + in the ambient space of symmetricmatrices. 3.113.2.0.0.1 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 [158,2.3.7]g(µY + (1 − µ)Z) ≼µg(Y ) + (1 − µ)g(Z) (522)S M +Reversing sense of the inequality flips this definition to concavity. Strictconvexity is defined less a stroke of the pen in (522) similarly to (423).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 (522). By dual generalized inequalities we have the equivalent butmore broad criterion, (2.13.5)g convex ⇔ 〈W , g〉 convex∀W ≽S M +0 (523)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. △3.11 Function symmetry is not a necessary requirement for convexity; indeed, for A∈R m×pand B ∈R m×k , g(X) = AX + B is a convex (affine) function in X on domain R p×k withrespect to the nonnegative orthant R m×k+ . Symmetric convex functions share the samebenefits as symmetric matrices. Horn & Johnson [150,7.7] liken symmetric matrices toreal numbers, and (symmetric) positive definite matrices to positive real numbers.