A Fast Gradient and Function Sampling Method for Finite Max-Functions

TL;DR

This paper introduces a novel sampling method for optimizing finite max-functions, demonstrating superlinear convergence properties and providing both theoretical analysis and illustrative examples.

Contribution

It proposes a new gradient and function sampling approach for nonsmooth, nonconvex finite max-functions, with proven superlinear convergence under certain conditions.

Findings

Superlinear convergence to minimizers

Improvement of optimality certificates

Theoretical convergence analysis

Abstract

This paper tackles the unconstrained minimization of a class of nonsmooth and nonconvex functions that can be written as finite max-functions. A gradient and function-based sampling method is proposed which, under special circumstances, either moves superlinearly to a minimizer of the problem of interest or superlinearly improves the optimality certificate. Global and local convergence analysis are presented, as well as illustrative examples that corroborate and elucidate the obtained theoretical results.