Volume 10 (2014) Article 4 pp. 77-105
Grothendieck Inequalities for Semidefinite Programs with Rank Constraint
Received: May 4, 2012
Revised: January 21, 2014
Published: May 31, 2014
Keywords: randomized rounding, Grothendieck inequality, $n$-vector model, XOR games
Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constraint is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than $1$ and we give two applications: approximating ground states in the $n$-vector model in statistical mechanics and XOR games in quantum information theory.