On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives

TL;DR

This paper develops a comprehensive theory for analyzing multifunctions using subspaces in generalized derivatives, leading to new calculus rules, a modified Newton method, and generalized inverse function theorems.

Contribution

It introduces a novel framework for local analysis of multifunctions via subspaces in generalized derivatives, enhancing existing calculus and stability results.

Findings

Constructed a modified semismooth* Newton method with better convergence.

Generalized Clarke's Inverse Function Theorem for multifunctions.

Provided new characterizations of strong metric regularity and tilt stability.

Abstract

The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the graphically Lipschitzian mappings and thus a number of multifunctions, frequently arising in optimization and equilibrium problems. The developed theory makes use of own generalized derivatives, provides us with some calculus rules and reveals a number of interesting connections. In particular, it enables us to construct a modification of the semismooth* Newton method with improved convergence properties and to derive a generalization of Clarke's Inverse Function Theorem to multifunctions together with new efficient characterizations of strong metric (sub)regularity and tilt stability.