On the exactness of Lasserre relaxations and pure states over real closed fields

TL;DR

This paper investigates conditions under which Lasserre relaxations precisely capture the convex hull of solution sets and optimal values in polynomial inequalities and optimization problems, especially for compact, outward-bulging sets.

Contribution

It establishes new criteria ensuring the exactness of Lasserre relaxations for polynomial systems and optimization, linking geometric properties of solution sets to relaxation accuracy.

Findings

Lasserre relaxations often exactly represent the convex hull for sufficiently high degree d.

Exactness of polynomial optimization relaxation depends on the geometry of the solution set and the properties of the objective function.

The paper provides bounds on d for which the relaxation matches the original problem's solution.

Abstract

Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most $d$. It defines a spectrahedron that projects down to a convex semialgebraic set containing $S$. In the best case, the projection equals the convex hull of $S$. We show that this is very often the case for sufficiently high $d$ if $S$ is compact and "bulges outwards" on the boundary of its convex hull. Now let additionally a polynomial objective function $f$ be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree $d$ is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if $S$ is compact and $d$ exceeds some bound that depends on the description of $S$ and certain characteristicae of $f$ like the mutual distance of its global minimizers on $S$.