Theta Bodies for Polynomial Ideals

Abstract

Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal. These relaxations generalize Lovász's construction of the theta body of a graph. We establish a relationship between theta bodies and Lasserre's relaxations for real varieties which allows, in many cases, for theta bodies to be expressed as feasible regions of semidefinite programs. Examples from combinatorial optimization are given. Lovász asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety. We answer this question for vanishing ideals of finite point sets via several equivalent characterizations. We also give a geometric description of the first theta body for all ideals.