On some generalized equations with metrically C-increasing mappings: solvability and error bounds with applications to optimization

TL;DR

This paper investigates the solvability and error bounds of generalized equations involving set-valued mappings and convex cones, introducing the metric C-increase property to ensure solutions and apply to optimization problems.

Contribution

It introduces the metric C-increase property for generalized equations, linking metric behavior to solution existence and error bounds, with applications in optimization and vector problems.

Findings

Metric C-increase guarantees solution existence.

Provides error bounds based on problem data.

Applications to optimization and vector solutions.

Abstract

Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential inclusions. The present paper contains a study on solvability and error bounds for generalized equations of the form $F(x)\subseteq C$, where $F$ is a given set-valued mapping and $C$ is a closed, convex cone. A property called metric $C$-increase, matching the metric behaviour of $F$ with the partial order associated with $C$, is singled out, which ensures solution existence and error bound estimates in terms of problem data. Applications to the exact penalization of optimization problems with constraint systems, defined by the above class of generalized equations, and to the existence of ideal efficient solutions in vector optimization are proposed.