On the convergence of critical points on real algebraic sets and applications to optimization

TL;DR

This paper studies the behavior of critical points on perturbed real algebraic sets, providing conditions for their existence and properties, with applications to optimization methods like interior point algorithms.

Contribution

It introduces new conditions ensuring the existence, finiteness, and convergence of critical points and central paths in polynomial and non-linear optimization.

Findings

Established conditions for critical point existence on perturbed sets.

Provided new insights into the convergence of central paths in polynomial optimization.

Extended convergence results to non-linear programs with definable sets.

Abstract

Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an infinitesimal perturbation $V_{\xi} = \zero(\{P_1-\xi_1,\ldots,P_s-\xi_s\},\R^n)$. Our main motivation is to understand the limiting behavior of local minimizers of the log-barrier function (and central paths) in polynomial optimization, whose existence plays a fundamental role, in theory and practice, for modern interior point methods. We establish different sets of conditions that ensure existence, finiteness, boundedness, and non-degeneracy of critical points of $F$ on $V_{\xi}$, respectively. These lead to new conditions for the existence, convergence, and smoothness of central paths of polynomial optimization and its extension to non-linear optimization problems involving definable sets and functions in an o-minimal structure. In particular, for non-linear programs defined by real globally analytic functions, our extension provides a stronger form of the convergence result obtained by Drummond and Peterzil.