Second-order analysis of piecewise linear functions with applications to optimization and stability

TL;DR

This paper develops second-order variational analysis for piecewise linear functions, providing new insights into optimization stability and conditions for local minimizers in complex problems.

Contribution

It introduces explicit second-order subdifferential calculations and links nondegeneracy with second-order qualification for composite functions.

Findings

Established relationships between nondegeneracy and second-order qualification.

Provided second-order characterization of stable local minimizers.

Applied analysis to optimization and stability issues.

Abstract

This paper is devoted to second-order variational analysis of a rather broad class of extended-real-valued piecewise liner functions and their applications to various issues of optimization and stability. Based on our recent explicit calculations of the second-order subdifferential for such functions, we establish relationships between nondegeneracy and second-order qualification for fully amenable compositions involving piecewise linear functions. We then provide a second-order characterization of full stable local minimizers in composite optimization and constrained minimax problems.