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Stability Criteria and Calculus Rules via Conic Contingent Coderivatives in Banach Spaces

Boris S. Mordukhovich, Pengcheng Wu, Xiaoqi Yang

Abstract

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolves around the newly introduced concept of $\varepsilon$-regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.

Stability Criteria and Calculus Rules via Conic Contingent Coderivatives in Banach Spaces

Abstract

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolves around the newly introduced concept of -regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.
Paper Structure (7 sections, 14 theorems, 178 equations)

This paper contains 7 sections, 14 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. $\varepsilon$-Regular Normal Cone and Preliminaries
  3. Neighborhood Conditions for Relative Lipschitz-like Multifunctions
  4. Pointbased Coderivative Characterizations of the Relative Lipschitz-like Property
  5. Chain Rules for Conic Contingent Coderivatives
  6. Relative Extremal Principle and Sum Rules for Conic Contingent Coderivatives
  7. Concluding Remarks

Key Result

Proposition 2.2

Let $X$ be a reflexive Banach space, let $\Omega\subset X$ be a closed and convex set, and let $x^*\in J(T(\bar{x};\Omega))$. Then for any $\varepsilon>0$, there exists $\delta>0$ such that we have

Theorems & Definitions (36)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 26 more