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21
Approximation Limits of Linear Programs (Beyond Hierarchies)
, 2013
"... We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generate ..."
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Cited by 17 (6 self)
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We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2−ɛ)approximations for CLIQUE require linear programs of size 2nΩ(ɛ). This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main technical ingredient is a quantitative improvement of Razborov’s rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.
Worstcase results for positive semidefinite rank. arXiv preprint arXiv:1305.4600
"... Abstract. This paper presents various worstcase results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic ndimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For ..."
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Cited by 6 (2 self)
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Abstract. This paper presents various worstcase results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic ndimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 dv/6e which in turn shows that the psd rank of a p × q matrix of rank three is at most 4 dmin{p, q}/6e. In general, a nonnegative matrix of rank (k+1 2 has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed. 1.
Positive semidefinite rank
, 2014
"... Let M ∈ Rp×q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices Ai, Bj of size k × k such that Mij = trace(AiBj). The psd rank has many appealing geometric interpretations, including semidefinite re ..."
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Cited by 6 (0 self)
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Let M ∈ Rp×q be a nonnegative matrix. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices Ai, Bj of size k × k such that Mij = trace(AiBj). The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and informationtheoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.
SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF ROTATION MATRICES
, 2015
"... We study the convex hull of SO(n), the set of n × n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive ..."
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Cited by 5 (3 self)
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We study the convex hull of SO(n), the set of n × n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.
Equivariant semidefinite lifts and sumofsquares hierarchies
, 2014
"... A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive s ..."
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Cited by 5 (2 self)
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A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive semidefinite lift (psd lift) of a polytope P is a representation of P as the projection of an affine slice of the positive semidefinite cone Sd+. Such a representation allows linear optimization problems over P to be written as semidefinite programs of size d. Such representations can be beneficial in practice when d is much smaller than the number of facets of the polytope P. In this paper we are concerned with socalled equivariant psd lifts (also known as symmetric psd lifts) which respect the symmetries of the polytope P. We present a representationtheoretic framework to study equivariant psd lifts of a certain class of symmetric polytopes known as regular orbitopes. Our main result is a structure theorem where we show that any equivariant psd lift of size d of a regular orbitope is of sumofsquares type where the functions in the sumofsquares decomposition come from an invariant subspace of dimension smaller than d2. We use this framework to study two wellknown families of polytopes, namely the parity polytope and the cut polytope, and we prove exponential lower bounds for any equivariant psd lifts of these polytopes.
Which nonnegative matrices are slack matrices?
, 2013
"... In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown. ..."
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Cited by 4 (0 self)
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In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.
On the Power of Symmetric LP and SDP Relaxations
"... We study the computational power of general symmetric relaxations for combinatorial optimization problems, both in the linear programming (LP) and semidefinite programming (SDP) case. We show new connections to explicit LP and SDP relaxations, like those obtained from standard hierarchies. Concrete ..."
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We study the computational power of general symmetric relaxations for combinatorial optimization problems, both in the linear programming (LP) and semidefinite programming (SDP) case. We show new connections to explicit LP and SDP relaxations, like those obtained from standard hierarchies. Concretely, for k < n/4, we show that krounds of sumofsquares / Lasserre relaxations of size k n