Convergence rate for linear minimizer-estimators in the moment-sum-of-squares hierarchy

TL;DR

This paper establishes convergence rates for measures obtained from the moment-sum-of-squares hierarchy in polynomial optimization, linking recent effective Positivstellensätze to practical convergence guarantees.

Contribution

It introduces a convergence rate analysis for measures in the moment-SoS hierarchy, extending understanding of their approach to optimal solutions in polynomial optimization.

Findings

Measures converge to probability measures on optimal points

Convergence rate transfers from effective Positivstellensätze

Analysis includes the upper bound hierarchy with convexity considerations

Abstract

Effective Positivstellens\"atze provide convergence rates for the moment-sum-of-squares (SoS) hierarchy for polynomial optimization (POP). In this paper, we add a qualitative property to the recent advances in those effective Positivstellens\"atze. We consider optimal solutions to the moment relaxations in the moment-SoS hierarchy and investigate the measures they converge to. It has been established that those limit measures are the probability measures on the set of optimal points of the underlying POP. We complement this result by showing that these measures are approached with a convergence rate that transfers from the (recent) effective Positivstellens\"atze. As a special case, this covers estimating the minimizer of the underlying POP via linear pseudo-moments. Finally, we analyze the same situation for another SoS hierarchy - the upper bound hierarchy - and show how convexity can be leveraged.