Projectional Coderivatives and Calculus Rules

TL;DR

This paper introduces projectional coderivatives and calculus rules in finite dimensions, providing refined fixed-point expressions and characterizations for smooth manifolds, enhancing analysis tools for variational problems.

Contribution

It develops a new tool, projectional coderivatives, with calculus rules and refined expressions for smooth manifolds, advancing variational analysis methods.

Findings

Projectional coderivatives can be expressed as fixed-point formulas on smooth manifolds.

The generalized Mordukhovich criterion is improved for better characterization.

Chain and sum rules are established for the new calculus framework.

Abstract

This paper is devoted to the study of a newly introduced tool, projectional coderivatives and the corresponding calculus rules in finite dimensions. We show that when the restricted set has some nice properties, more specifically, is a smooth manifold, the projectional coderivative can be refined as a fixed-point expression. We will also improve the generalized Mordukhovich criterion to give a complete characterization of the relative Lipschitz-like property under such a setting. Chain rules and sum rules are obtained to facilitate the application of the tool to a wider range of problems.