Convergence rate for linear minimizer-estimators in the moment-sum-of-squares hierarchy
Corbinian Schlosser
TL;DR
This work advances the theory of the moment-SOS hierarchy for polynomial optimization by providing a quantitative convergence rate for the moment-relaxation solutions to the measures supported on the set of POP minimizers. Building on effective Putinar-type bounds and Łojasiewicz-type arguments, it shows that the pseudo-moments generated by moment relaxations approximate the moments of optimal measures at a polynomial rate, and in the case of a unique minimizer, enables a polynomial-rate estimator for the minimizer via linear moments. The analysis extends to an upper-bound SOS hierarchy, establishing how convexity properties yield feasible estimators and a priori cost bounds. By connecting convergence of objective values to the convergence of minimizers, the results enhance the practical reliability of SOS-based methods for POPs and clarify how improvements in Positivstellensätze translate into stronger convergence guarantees.
Abstract
Effective Positivstellensätze 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 Positivstellensätze. 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 Positivstellensätze. 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.