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The strength of pseudo-expectations : a detailed analysis of the work of Lee, Raghavendra and Steurer on the psd rank of the family of correlation polytopes

Sebastian Gruler

Published Jan 1, 2018• 0 citations

Abstract

In combinatorial optimization, many problems can be modeled by optimizing a linear functional over a polytope. The standard tool to manage such a problem is the well-known linear programming. But on this occasion, linear programming reaches its limits for example for the family of the correlation polytopes CORRn (n ϵ IN), that forms a very important family of polytopes in combinatorial optimization, because the number of facets of any polytope that maps linearly onto CORRn grows exponentially in n. The question rises whether semidefinite programming, a broad generalization of linear programming, can provide relief. The feasible set of a semidefinite program (SDP) is a so-called spectrahedron, the solution set of a linear matrix inequality (LMI). Instead of the number of facets of the polytope in the case of a linear program, the size of the LMI defining the spectrahedron is a measure for the complexity of the SDP. In their breakthrough work, Lee, Raghavendra and Steurer [LRS] negated 2015 this question. They proved a super-polynomial lower bound in n on the size of any LMI defining any spectrahedron that linearly maps onto CORRn

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In combinatorial optimization, many problems can be modeled by optimizing a linear functional over a polytope. The standard tool to manage such a problem is the well-known linear programming. But on this occasion, linear programming reaches its limits for example for the family of the correlation polytopes CORRn (n ϵ IN), that forms a very important family of polytopes in combinatorial optimization, because the number of facets of any polytope that maps linearly onto CORRn grows exponentially in n. The question rises whether semidefinite programming, a broad generalization of linear programming, can provide relief. The feasible set of a semidefinite program (SDP) is a so-called spectrahedron, the solution set of a linear matrix inequality (LMI). Instead of the number of facets of the polytope in the case of a linear program, the size of the LMI defining the spectrahedron is a measure for the complexity of the SDP. In their breakthrough work, Lee, Raghavendra and Steurer [LRS] negated 2015 this question. They proved a super-polynomial lower bound in n on the size of any LMI defining any spectrahedron that linearly maps onto CORRn

There has been a lot of interest recently in proving lower bounds on the size of linear programs needed to represent a given polytope P. In a breakthrough paper Fiorini et al. [Proceedings of 44th ACM Symposium on Theory of Computing 2012, pages 95-106] showed that any linear programming formulation of maximum-cut must have exponential size. A natural question to ask is whether one can prove such strong lower bounds for semidefinite programming formulations. In this paper we take a step towards this goal and we prove strong lower bounds for a certain class of SDP formulations, namely SDPs over the Cartesian product of fixed-size positive semidefinite cones. In practice this corresponds to semidefinite programs with a block-diagonal structure and where blocks have constant size d. We show that any such extended formulation of the cut polytope must have exponential size (when d is fixed). The result of Fiorini et al. for LP formulations is obtained as a special case when d=1. For blocks of size d=2 the result rules out any small formulations using second-order cone programming. Our study of SDP lifts over Cartesian product of fixed-size positive semidefinite cones is motivated mainly from practical considerations where it is well known that such SDPs can be solved more efficiently than general SDPs. The proof of our lower bound relies on new results about the sparsity pattern of certain matrices with small psd rank, combined with an induction argument inspired from the recent paper by Kaibel and Weltge [arXiv:1307.3543] on the LP extension complexity of the correlation polytope.

It is well known that state-of-the-art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes. This leads us to consider the question, how well can we approximate semidefinite programs with linear programs? In this paper, we prove lower bounds on the size of linear programs that approximate the positive semidefinite cone. Let D be the set of n × n positive semidefinite matrices of trace equal to one. We prove two results on the hardness of approximating D with polytopes. We show that if 0 < ε < 1and A is an arbitrary matrix of trace equal to one, any polytope P such that (1-ε) (D-A) ⊂ P ⊂ D-A must have linear programming extension complexity at least [Formula: see text], where c > 0 is a constant that depends on ε. Second, we show that any polytope P such that D ⊂ P and such that the Gaussian width of P is at most twice the Gaussian width of D must have extension complexity at least [Formula: see text]. Our bounds are both superpolynomial in n and demonstrate that there is no generic way of approximating semidefinite programs with compact linear programs. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.

Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies.