# On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

## TL;DR

This paper proves that for certain convex semialgebraic sets satisfying specific polynomial conditions, the Lasserre relaxation is exact, simplifying the process of obtaining semidefinite representations.

## Contribution

The authors demonstrate that under natural convexity and second order quasiconcavity conditions, a single Lasserre relaxation suffices for exact representation, improving upon previous non-constructive methods.

## Key findings

- Lasserre relaxation is exact for the specified convex sets.
- The approach simplifies previous semidefinite representation methods.
- Conditions include second order strict quasiconcavity and the Archimedean property.

## Abstract

Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. In their seminal work of 2009, Helton and Nie showed under the same conditions that $S$ is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a non-constructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of $S$.