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The Gram dimension of a graph
 In Proceedings of the 2nd International Symposium on Combinatorial Optimization (A.R. Mahjoub et al., Eds.): ISCO 2012, LNCS 7422
, 2012
"... Abstract. The Gram dimension gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in Rk, having the same inner products on the edges of the graph. The class of graphs satisfying ..."
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Abstract. The Gram dimension gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in Rk, having the same inner products on the edges of the graph. The class of graphs satisfying gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is Kk+1. We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of drealizability of graphs. In particular, our result implies the characterization of 3realizable graphs of Belk and Connelly [5,6]. 1
Lowrank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems
 in Conference on Decision and Control
, 2014
"... Abstract—This paper is concerned with the problem of finding a lowrank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimumrank solution of the LMI problem to the spars ..."
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Cited by 4 (3 self)
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Abstract—This paper is concerned with the problem of finding a lowrank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimumrank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose two graphtheoretic convex programs to obtain a lowrank solution. The first convex optimization needs a tree decomposition of the sparsity graph. The second one does not rely on any computationallyexpensive graph analysis and is always polynomialtime solvable. The results of this work can be readily applied to three separate problems of minimumrank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks. I.
Finding lowrank solutions of sparse linear matrix inequalities using convex optimization
 Online]. Available: http://www.ee.columbia.edu/lavaei/LMI Low Rank.pdf
, 2014
"... Abstract. This paper is concerned with the problem of finding a lowrank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimumrank solution of the LMI problem to the spar ..."
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Cited by 4 (4 self)
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Abstract. This paper is concerned with the problem of finding a lowrank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimumrank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose three graphtheoretic convex programs to obtain a lowrank solution. Two of these convex optimization problems need a tree decomposition of the sparsity graph, which is an NPhard problem in the worst case. The third one does not rely on any computationallyexpensive graph analysis and is always polynomialtime solvable. The results of this work can be readily applied to three separate problems of minimumrank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks.
Positive Semidefinite Matrix Completion, Universal Rigidity and the Strong Arnold Property
, 2013
"... This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a su ..."
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Cited by 4 (1 self)
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This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly’s sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension gd(·) and the Colin de Verdière type graph parameter ν =(·).
Complexity of the positive semidefinite matrix completion problem with a rank constraint
, 2014
"... We consider the decision problem asking whether a partial rational symmetric matrix with an allones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is NPhard for any fixed integer k ≥ 2. Equivalently, for k ≥ 2, it is NPhard to test me ..."
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Cited by 3 (3 self)
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We consider the decision problem asking whether a partial rational symmetric matrix with an allones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is NPhard for any fixed integer k ≥ 2. Equivalently, for k ≥ 2, it is NPhard to test membership in the rank constrained elliptope Ek(G), i.e., the set of all partial matrices with offdiagonal entries specified at the edges of G, that can be completed to a positive semidefinite matrix of rank at most k. Additionally, we show that deciding membership in the convex hull of Ek(G) is also NPhard for any fixed integer k ≥ 2.
Complexity of the positive semidefinite matrix completion problem with a . . .
, 2012
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GraphTheoretic Algorithms for Polynomial Optimization Problems
"... Abstract — The objective of this tutorial paper is to study a general polynomial optimization problem using a semidefinite programming (SDP) relaxation. The first goal is to show how the underlying structure and sparsity of an optimization problem affect its computational complexity. Graphtheoretic ..."
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Abstract — The objective of this tutorial paper is to study a general polynomial optimization problem using a semidefinite programming (SDP) relaxation. The first goal is to show how the underlying structure and sparsity of an optimization problem affect its computational complexity. Graphtheoretic algorithms are presented to address this problem based on the notions of lowrank optimization and matrix completion. By building on this result, it is then shown that every polynomial optimization problem admits a sparse representation whose SDP relaxation has a rank 1 or 2 solution. The implications of these results are discussed in details and their applications in decentralized control and power systems are also studied. I.
Selected Open Problems in Discrete Geometry and Optimization
, 2013
"... A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will ..."
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A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers.