v2010.10.26 - Convex Optimization

266 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSand all X ∈ R n×N : Consider Schur-form (1514) fromA.4: for T ∈ S n[ ]A + ǫI XTG(A, X, T ) =X ǫ −1 ≽ 0T⇔T − ǫX(A + ǫI) −1 X T ≽ 0A + ǫI ≻ 0(633)By Theorem 2.1.9.0.1, inverse image of the positive semidefinite cone S N+n+under affine mapping G(A, X, T ) is convex. Function g(A, X) is convexon S N + ×R n×N because its epigraph is that inverse image:epig(A, X) = { (A, X, T ) | A + ǫI ≻ 0, ǫX(A + ǫI) −1 X T ≼ T } = G −1( S N+n+(634)3.7.3 second-order convexity condition, matrix-valued fThe following line theorem is a potent tool for establishing convexity of amultidimensional function. To understand it, what is meant by line must firstbe solidified. Given a function g(X) : R p×k →S M and particular X, Y ∈ R p×knot necessarily in that function’s domain, then we say a line {X+ t Y | t ∈ R}passes through domg when X+ t Y ∈ domg over some interval of t ∈ R .3.7.3.0.1 Theorem. Line theorem. [61,3.1.1]Matrix-valued function g(X) : R p×k →S M is convex in X if and only if itremains convex on the intersection of any line with its domain. ⋄Now we assume a twice differentiable function.3.7.3.0.2 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 (635))