v2010.10.26 - Convex Optimization

268 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSand X is skinny-or-square full-rank (Corollary A.3.1.0.5).‖X‖ 2 sup ‖Xa‖ 2 = σ(X) 1 = √ λ(X T X) 1 = minimize‖a‖=1t∈Rsubject tot (639)[ ] tI XX T ≽ 0tIThe matrix 2-norm (spectral norm) coincides with largest singular value.This supremum of a family of convex functions in X must be convex becauseit constitutes an intersection of epigraphs of convex functions.3.7.3.0.5 Exercise. Squared maps.Give seven examples of distinct polyhedra P for which the set{X T X | X ∈ P} ⊆ S n + (640)were convex. Is this set convex, in general, for any polyhedron P ?(confer (1233) (1240)) Is the epigraph of function g(X)=X T X convex forany polyhedral domain?3.7.3.0.6 Exercise. Squared inverse. (confer3.7.2.0.1)For positive scalar a , real function f(x)= ax −2 is convex on the nonnegativereal line. Given positive definite matrix constant A , prove via line theoremthat g(X)= tr ( (X T A −1 X) −1) is generally not convex unless X ≻ 0 . 3.26From this result, show how it follows via Definition 3.7.0.0.1-2 thath(X) = (X T A −1 X) −1 is generally neither convex.3.7.3.0.7 Example. Matrix exponential.The matrix-valued function g(X) = e X : S M → S M is convex on the subspaceof circulant [168] symmetric matrices. Applying the line theorem, for all t∈Rand circulant X, Y ∈ S M , from Table D.2.7 we haved 2dt 2eX+ t Y = Y e X+ t Y Y ≽S M +0 , (XY ) T = XY (641)because all circulant matrices are commutative and, for symmetric matrices,XY = Y X ⇔ (XY ) T = XY (1474). Given symmetric argument, the matrix3.26 Hint:D.2.3.