v2010.10.26 - Convex Optimization

3.7. CONVEX MATRIX-VALUED FUNCTION 267Similarly, ifd 2dt 2 g(X+ t Y ) ≻ S M +0 (636)then g is strictly convex; the converse is generally false. [61,3.1.4] 3.23 △3.7.3.0.3 Example. Matrix inverse. (confer3.3.1)The matrix-valued function X µ is convex on int S M + for −1≤µ≤0or 1≤µ≤2 and concave for 0≤µ≤1. [61,3.6.2] In particular, thefunction g(X) = X −1 is convex on int S M + . For each and every Y ∈ S M(D.2.1,A.3.1.0.5)d 2dt 2 g(X+ t Y ) = 2(X+ t Y )−1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 ≽S M +0 (637)on some open interval of t ∈ R such that X + t Y ≻0. Hence, g(X) isconvex in X . This result is extensible; 3.24 trX −1 is convex on that samedomain. [202,7.6 prob.2] [55,3.1 exer.25]3.7.3.0.4 Example. Matrix squared.Iconic real function f(x)= x 2 is strictly convex on R . The matrix-valuedfunction g(X)=X 2 is convex on the domain of symmetric matrices; forX, Y ∈ S M and any open interval of t ∈ R (D.2.1)d 2d2g(X+ t Y ) =dt2 dt 2(X+ t Y )2 = 2Y 2 (638)which is positive semidefinite when Y is symmetric because then Y 2 = Y T Y(1457). 3.25A more appropriate matrix-valued counterpart for f is g(X)=X T Xwhich is a convex function on domain {X ∈ R m×n } , and strictly convexwhenever X is skinny-or-square full-rank. This matrix-valued function canbe generalized to g(X)=X T AX which is convex whenever matrix A ispositive semidefinite (p.692), and strictly convex when A is positive definite3.23 The strict-case converse is reliably true for quadratic forms.3.24 d/dt tr g(X+ tY ) = trd/dtg(X+ tY ). [203, p.491]3.25 By (1475) inA.3.1, changing the domain instead to all symmetric and nonsymmetricpositive semidefinite matrices, for example, will not produce a convex function.