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Subdifferential Calculus for Ordered Set-Valued Mappings between Infinite-Dimensional Spaces

Boris S. Mordukhovich, Oanh Nguyen

Abstract

The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various constraints in infinite dimensions. The main results establish new sum and chain rules for major subdifferential constructions associated with ordered set-valued mappings under appropriate qualification and sequentially normal compactness conditions.

Subdifferential Calculus for Ordered Set-Valued Mappings between Infinite-Dimensional Spaces

Abstract

The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various constraints in infinite dimensions. The main results establish new sum and chain rules for major subdifferential constructions associated with ordered set-valued mappings under appropriate qualification and sequentially normal compactness conditions.

Paper Structure

This paper contains 4 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Machinery of Variational Analysis
  3. Ordered Set-Valued Mappings and Their Subdifferentials
  4. Sudifferential Sum Rules for Ordered Set-Valued Mappings

Key Result

Theorem 2.1

Let $F \colon X \rightrightarrows Z$ be a set-valued mapping between Asplund spaces, and let the graph of $F$ be closed around $(\bar{x}, \bar{z}) \in \hbox{\rm gph}\, F$. Then the following properties are equivalent:

Theorems & Definitions (10)

  • Theorem 2.1: characterization of Lipschitz-like multifunctions
  • Definition 3.1: subdifferentials of ordered set-valued mappings
  • Lemma 4.1: coderivative sum rules
  • Lemma 4.2: epigraphical multifunctions under set summation
  • Theorem 4.3: sum rules for subdifferentials
  • Corollary 4.4: subdifferential sum rules for ELL mappings
  • Lemma 4.5: singular subdifferentials of the indicator mappings
  • Lemma 4.6: regular subdifferentials of the restricted multifunctions
  • Theorem 4.7: subdifferentials of restricted multifunctions
  • Example 4.8: illustration of the subdifferential sum rules