Convex Hulls of Algebraic Sets

TL;DR

This paper introduces a method for approximating the convex hulls of algebraic sets using sums of squares and moment matrices, extending Lasserre's hierarchy with applications to graph theta bodies.

Contribution

It develops a novel technique for convexifying algebraic sets via sums of squares, with explicit convergence analysis and applications to real radical ideals.

Findings

Method computes convex hulls using polynomial relaxations.

Finite convergence is explicitly characterized for finite point sets.

Applications include extensions of Lovász's theta body for graphs.

Abstract

This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lov\'asz concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic set in R^n. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described explicitly for finite point sets.