$\alpha $-Lower Subdifferentiable Functions

Abstract

In this paper the authors introduce the notion of $\alpha $-lower subdifferentiability, with $\alpha \in ( 0,1 ]$, for extended real-valued functions defined on a locally convex real topological vector space X. This is a generalization of the concept of lower subdifferentiability due to Plastria, which corresponds to the case $\alpha = 1$. When X is a normed space, the class of $\alpha $-lower subdifferentiable functions appears to be closely related to that of $\alpha $-Hölder quasi-convex functions. Two applications to quasi-convex optimization are given: a duality theorem, based on conjugation with respect to $h_\alpha $, and Kuhn–Tucker–type optimality conditions in terms of $\alpha $-lower subdifferentials.