Strong and Weak Convexity of Sets and Functions

TLDR

Strong or weak convexity of a function is determined by the sign of the parameter ρ. The paper studies strongly and weakly convex sets. The authors derive properties of these sets using supporting balls and normal cones, analyze ρ‑convex functions, and relate them to their epigraphs and level sets. They provide three applications: a separation theorem, a sufficient condition for global optimality in a nonconvex program, and a geometric condition for a set to be a manifold.

Abstract

In this paper we study two classes of sets, strongly and weakly convex sets. For each class we derive a series of properties which involve either the concept of supporting ball, an obvious extension of the concept of supporting hyperplane, or the normal cone to the set. We also study a class of functions, denoted ρ-convex, which satisfy for arbitrary points x 1 and x 2 and any value λ ∈ [0, 1] the classical inequality of convex functions up to a term ρ(1 − λ) λ‖x 1 − x 2 ‖ 2 . Depending on the sign of the constant ρ the function is said to be strongly or weakly convex. We provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets. Finally, we give three applications: a separation theorem, a sufficient condition for global optimum of a nonconvex programming problem, and a sufficient geometrical condition for a set to be a manifold.