v2010.10.26 - Convex Optimization

270 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONSFigure 79: Iconic unimodal differentiable quasiconvex function of twovariables graphed in R 2 × R on some open disc in R 2 . Note reversal ofcurvature in direction of gradient.Unlike convex functions, quasiconvex functions are not necessarilycontinuous; e.g., quasiconcave rank(X) on S M + (2.9.2.9.2) and card(x)on R M + . Although insufficient for convex functions, convexity of each andevery sublevel set serves as a definition of quasiconvexity:3.8.0.0.2 Definition. Quasiconvex multidimensional function.Scalar-, vector-, or matrix-valued function g(X) : R p×k →S M is a quasiconvexfunction of matrix X iff domg is a convex set and the sublevel setcorresponding to each and every S ∈ S ML Sg = {X ∈ dom g | g(X) ≼ S } ⊆ R p×k (631)is convex. Vectors are compared with respect to the nonnegative orthant R M +while matrices are with respect to the positive semidefinite cone S M + .Convexity of the superlevel set corresponding to each and every S ∈ S M ,likewiseL S g = {X ∈ domg | g(X) ≽ S } ⊆ R p×k (644)is necessary and sufficient for quasiconcavity.△