Strictly and Roughly Convexlike Functions

Abstract

A function $f:D \subseteq \mathbb{R}^n \to \mathbb{R}$ is said to be strictly and roughly convexlike with respect to the roughness degree r > 0 (for short, strictly r-convexlike) provided that, for all x ₀, x ₁ ∈ D satisfying ||x ₀ − x ₁|| > r, there exists a λ ∈ ]0, 1[ such that

$$f((1 - \lambda )x_0 + \lambda x_1 ) < (1 - \lambda )f(x_0 ) + \lambda f(x_1 ).$$

.

The most important property of strictly r-convexlike functions is that the diameter of the set of global minimizers is not greater than r. This property is needed in another paper for obtaining the rough stability of optimal solutions to nonconvex parametric optimization problems. Moreover, if f is supposed to be lower semicontinuous, then each r-local minimizer x*, defined by

$$f(x*) \leqslant f(x),{\text{ for all }}x \in D{\text{ with }}\left\| {x - x*} \right\| \leqslant r,$$

is a global minimizer of f. In this paper, necessary and sufficient conditions for a function to be strictly r-convexlike are stated. In particular, the class of strictly γ -convex functions is considered.