On invex functions with same η in single and multivalued nonsmooth optimization with Clarke's subdifferential

TL;DR

The paper addresses characterizing families of nonsmooth locally Lipschitz continuous functions that are invex with respect to a common function $\boldsymbol \eta$ in both single- and multivalued optimization. It develops a Clarke-subdifferential framework to relate invexity to scalarized forms, demonstrates that a naive generalization from differentiable settings can fail without Clarke regularity, and proves an equivalence theorem linking V-invexity with the same $\boldsymbol \eta$ to invexity of all scalarizations. A counterexample clarifies the necessity of Clarke regularity, while the main result (an equivalence among four conditions) provides a solid theoretical foundation for optimality conditions and algorithm design in nonsmooth multiobjective settings. The work lays groundwork for future exploration of alternative nonsmooth tools, such as Dem or quasidifferential methods, to extend invexity concepts beyond Clarke subdifferentials.

Abstract

In this paper, a finite family of nonsmooth locally Lipschitz continuous functions that are invex with respect to the same function η are characterized in terms of their scalarized counterparts.

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Definition 2.7
• Definition 3.1
• Definition 3.2