Subdifferentials at infinity and applications in optimization

TL;DR

This paper introduces new concepts of subdifferentials and normal cones at infinity, providing calculus rules and characterizations that enhance understanding of optimization problems involving unbounded sets and functions.

Contribution

It develops the theory of subdifferentials at infinity and applies these concepts to optimize conditions, stability, and properties of unbounded functions.

Findings

Characterization of Lipschitz continuity at infinity.

New calculus rules for subdifferentials at infinity.

Applications to optimality conditions and stability in optimization.

Abstract

In this work, the notions of normal cones at infinity to unbounded sets and limiting and singular subdifferentials at infinity for extended real value functions are introduced. Various calculus rules for these notions objects are established. A complete characterization of the Lipschitz continuity at infinity for lower semi-continuous functions is given. The obtained results are aimed ultimately at applications to diverse problems of optimization, such as optimality conditions, coercive properties, weak sharp minima and stability results.