Functions Which Are Quasiconvex under Linear Perturbations
Emmanuel Nicholas Barron, Rafal K. Goebel, Robert R. Jensen · SIAM Journal on Optimization · 2012
A quasiconvex function is a function which has convex sublevel sets. This paper studies robustly quasiconvex functions, that is, quasiconvex functions which remain quasiconvex under small linear perturbations. Relations to pseudoconvexity and other generalized convexity concepts and necessary and sufficient first-order conditions for robust quasiconvexity of smooth functions are presented. Convex-analytic properties and convexification of robustly quasiconvex functions are studied. Supporting robustly quasiconvex functions by simpler functions is discussed, with duality theory as motivation. Through the use of a second-order condition for robust quasiconvexity of nonsmooth functions, nontrivial examples of such functions are given.