Coincidence Points of Parameterized Generalized Equations with Applications to Optimal Value Functions

TL;DR

This paper develops a comprehensive framework for analyzing coincidence points of parameterized set-valued mappings, providing new theorems and conditions that enhance understanding of solution existence and stability in variational analysis and optimization.

Contribution

It introduces a general theorem for coincidence points with explicit error bounds, leading to new implicit function theorems and conditions for stability of optimal value functions.

Findings

Established a general theorem for coincidence points with error bounds

Derived a new implicit function theorem for parameterized equations

Provided conditions for semicontinuity and continuity of optimal value functions

Abstract

The paper studies coincidence points of parameterized set-valued mappings (multifunctions), which provide an extended framework to cover several important topics in variational analysis and optimization that include the existence of solutions of parameterized generalized equations, implicit function and fixed-point theorems, optimal value functions in parametric optimization, etc. Using the advanced machinery of variational analysis and generalized differentiation that furnishes complete characterizations of well-posedness properties of multifunctions, we establish a general theorem ensuring the existence of parameter-dependent coincidence point mappings with explicit error bounds for parameterized multifunctions between infinite-dimensional spaces. The obtained major result yields a new implicit function theorem and allows us to derive efficient conditions for semicontinuity and continuity of optimal value functions associated with parametric minimization problems subject to constraints governed by parameterized generalized equations.