Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates

TL;DR

This paper develops new error bounds for parametric polynomial systems, extending the Łojasiewicz inequality, and applies these results to stability analysis and convergence rates in various polynomial optimization problems.

Contribution

It extends the Łojasiewicz gradient inequality to nonsmooth supremum marginal functions and derives higher-order local error bounds with explicit exponents.

Findings

Established generalized Łojasiewicz inequalities for polynomial systems.

Derived explicit convergence rates for cyclic projection algorithms.

Provided stability estimates for polynomial optimization problems.

Abstract

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (H\"older type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.