Clarke's tangent cones, subgradients, optimality conditions and the Lipschitzness at infinity

TL;DR

This paper investigates Clarke's tangent cones at infinity, subgradients, and optimality conditions for unbounded sets and functions, providing new characterizations and rules for computing subgradients and Lipschitz continuity at infinity.

Contribution

It introduces new properties of Clarke's tangent cones at infinity, characterizes subgradients at infinity, and establishes optimality conditions and Lipschitzness criteria at infinity.

Findings

Clarke's tangent cones at infinity are closed convex.

Characterization of the interior of tangent cones at infinity.

Necessary optimality conditions at infinity for optimization problems.

Abstract

We first study Clarke's tangent cones at infinity to unbounded subsets of $\mathbb{R}^n.$ We prove that these cones are closed convex and show a characterization of their interiors. We then study subgradients at infinity for extended real value functions on $\mathbb{R}^n$ and derive necessary optimality conditions at infinity for optimization problems. We also give a number of rules for the computing of subgradients at infinity and provide some characterizations of the Lipschitz continuity at infinity for lower semi-continuous functions.