v2007.09.17 - Convex Optimization

218 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS3.2.3.0.2 Definition. Differentiable convex matrix-valued function.Matrix-valued function g(X) : R p×k →S M is convex in X iff domg is anopen convex set, and its second derivative g ′′ (X+ t Y ) : R→S M is positivesemidefinite on each point of intersection along every line {X+ t Y | t ∈ R}that intersects domg ; id est, iff for each and every X, Y ∈ R p×k such thatX+ t Y ∈ domg over some open interval of t ∈ Rd 2dt 2 g(X+ t Y ) ≽ S M +0 (533)Similarly, ifd 2dt 2 g(X+ t Y ) ≻ S M +0 (534)then g is strictly convex; the converse is generally false. [46,3.1.4] 3.11 △3.2.3.0.3 Example. Matrix inverse. (confer3.1.5)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. [46,3.6.2] 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 (535)on some open interval of t ∈ R such that X + t Y ≻0. Hence, g(X) isconvex in X . This result is extensible; 3.12 trX −1 is convex on that samedomain. [150,7.6, prob.2] [41,3.1, exer.25]3.2.3.0.4 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)3.11 Quadratic forms constitute a notable exception where the strict-case converse isreliably true.3.12 d/dt trg(X+ tY ) = trd/dt g(X+ tY ). [151, p.491]