Matrix analysis
Theory of linear and integer programming
Theory of linear and integer programming
Proceedings of the first Malta conference on Graphs and combinatorics
SIAM Review
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
An Explicit Equivalent Positive Semidefinite Program for Nonlinear 0-1 Programs
SIAM Journal on Optimization
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
Semidefinite representations for finite varieties
Mathematical Programming: Series A and B
Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals
Foundations of Computational Mathematics
Linear Level Lasserre Lower Bounds for Certain k-CSPs
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Convex sets with semidefinite representation
Mathematical Programming: Series A and B
A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs
Mathematical Programming: Series A and B
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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.