Approximate volume and integration for basic semi-algebraic sets

TL;DR

This paper presents a hierarchy-based method using semidefinite and linear programming to accurately approximate the volume, moments, and integrals over basic semi-algebraic sets, with practical numerical considerations.

Contribution

It introduces a novel hierarchy of optimization problems that converges to the volume and moments of semi-algebraic sets, enabling precise integration approximation.

Findings

Converging sequence of volume estimates for semi-algebraic sets.

Approximation of moments and integrals of polynomials on these sets.

Discussion of numerical issues in the algorithms.

Abstract

Given a basic compact semi-algebraic set $\K\subset\R^n$, we introduce a methodology that generates a sequence converging to the volume of $\K$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on $\K$ can be approximated as closely as desired, and so permits to approximate the integral on $\K$ of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.