v2007.09.13 - Convex Optimization

3.3. QUASICONVEX 221A quasiconcave function is determined:f(µ Y + (1 − µ)Z) ≥ min{f(Y ), f(Z)} (539)Unlike convex functions, quasiconvex functions are not necessarilycontinuous; e.g., quasiconcave rank(X) on S M + (2.9.2.6.2) and card(x)on R M + . Although insufficient for convex functions, convexity of each andevery sublevel set serves as a definition of quasiconvexity:3.3.0.0.2 Definition. Quasiconvex multidimensional function.Scalar-, vector-, or matrix-valued function g(X) : R p×k →S M is a quasiconvexfunction of matrix X iff dom g is a convex set and the sublevel setcorresponding to each and every S ∈ S ML Sg = {X ∈ domg | g(X) ≼ S } ⊆ R p×k (529)△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 (540)is necessary and sufficient for quasiconcavity.△3.3.0.0.3 Exercise. Nonconvexity of matrix product.Consider the real function on a positive definite domain[f(X) = tr(X 1 X 2 ) , domf =∆ rel int SN+ 00 rel int S N +](541)wherewith superlevel setsX ∆ =[ ]X1 00 X 2≻S 2N+0 (542)L s f = {X ∈ domf | f(X) ≥ s }= {X ∈ domf | 〈X 1 , X 2 〉 ≥ s }(543)