v2010.10.26 - Convex Optimization

3.7. CONVEX MATRIX-VALUED FUNCTION 2633.7.0.0.1 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 [215,2.3.7]g(µY + (1 − µ)Z) ≼µg(Y ) + (1 − µ)g(Z) (621)S M +Reversing sense of the inequality flips this definition to concavity. Strictconvexity is defined less a stroke of the pen in (621) similarly to (500).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 (621). By dual generalized inequalities we have the equivalent butmore broad criterion, (2.13.5)g convex w.r.t S M + ⇔ 〈W , g〉 convexfor each and every W ≽S M∗+0 (622)Strict convexity on both sides requires caveat W ≠ 0. Because theset of all extreme directions for the selfdual positive semidefinite cone(2.9.2.7) comprises a minimal set of generators for that cone, discretization(2.13.4.2.1) allows replacement of matrix W with symmetric dyad ww T asproposed.△3.7.0.0.2 Example. Taxicab distance matrix.Consider an n-dimensional vector space R n with metric induced by the1-norm. Then distance between points x 1 and x 2 is the norm of theirdifference: ‖x 1 −x 2 ‖ 1 . Given a list of points arranged columnar in a matrixX = [x 1 · · · x N ] ∈ R n×N (76)respect to the nonnegative orthant R m×k+ . Symmetric convex functions share the samebenefits as symmetric matrices. Horn & Johnson [202,7.7] liken symmetric matrices toreal numbers, and (symmetric) positive definite matrices to positive real numbers.