Conservative parametric optimality and the ridge method for tame min-max problems

TL;DR

This paper investigates the convergence of the ridge method for min-max problems without convexity or smoothness assumptions, establishing conditions under which the method converges to equilibria satisfying optimality.

Contribution

It introduces a new characterization of definable conservative fields and proves the convergence of the ridge method in definable contexts, extending understanding of nonsmooth optimization.

Findings

Ridge method converges to equilibria in definable settings.

Definability ensures the parametric optimality formula is a conservative field.

Conservativity may fail for non-definable nonsmooth functions, leading to poor behavior.

Abstract

We study the ridge method for min-max problems, and investigate its convergence without any convexity, differentiability or qualification assumption. The central issue is to determine whether the ''parametric optimality formula'' provides a conservative field, a notion of generalized derivative well suited for optimization. The answer to this question is positive in a semi-algebraic, and more generally definable, context. The proof involves a new characterization of definable conservative fields which is of independent interest. As a consequence, the ridge method applied to definable objectives is proved to have a minimizing behavior and to converge to a set of equilibria which satisfy an optimality condition. Definability is key to our proof: we show that for a more general class of nonsmooth functions, conservativity of the parametric optimality formula may fail, resulting in an absurd behavior of the ridge method.