Error Bounds for Polynomial Optimization over the Hypercube using Putinar type Representations

TL;DR

This paper derives explicit degree and error bounds for polynomial optimization over the hypercube using Putinar's Positivstellensatz, employing Bernstein approximation techniques to improve positivity certificates.

Contribution

It introduces new bounds for Putinar type representations by connecting quadratic modules and preorderings for hypercube polynomials.

Findings

Provides explicit error bounds for polynomial approximations.

Relates quadratic modules and preorderings for hypercube constraints.

Uses Bernstein approximation to improve positivity certificates.

Abstract

Consider the optimization problem $p_{\min, Q} := \min_{\mathbf{x} \in Q} p(\mathbf{x})$, where $p$ is a degree $m$ multivariate polynomial and $Q := [0, 1]^n$ is the hypercube. We provide explicit degree and error bounds for the sums of squares approximations of $p_{\min, Q}$ corresponding to the Positivstellensatz of Putinar. Our approach uses Bernstein multivariate approximation of polynomials, following the methodology of De Klerk and Laurent to provide error bounds for Schm\"udgen type positivity certificates over the hypercube. We give new bounds for Putinar type representations by relating the quadratic module and the preordering associated with the polynomials $g_i := x_i (1 - x_i), \: i=1,\dots,n$, describing the hypercube $Q$.