Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization

Madani, Ramtin; Sojoudi, Somayeh; Fazelnia, Ghazal; Lavaei, Javad

doi:10.1137/14099379x

Cited by 31 publications

(17 citation statements)

References 53 publications

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“…So, for example when G is a tree, r * ≤ 2. Similar approaches and extensions can be found in [27,19,20]. In fact, [19] proves that any polynomial optimization problem can be reformulated as a QCQP with a corresponding SDP relaxation having r * ≤ 2.…”

Section: Rank Boundsmentioning

confidence: 81%

Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Burer

2019

Math. Program.

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We study a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, which contain no off-diagonal terms x j x k for j = k, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-timecheckable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

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Section: Rank Boundsmentioning

confidence: 81%

Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Burer

2019

Math. Program.

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“…DefineX opt (n) as the n × n principal submatrix of an optimal solution of (4). It is shown in [80] that:…”

Section: Matrix Completionmentioning

confidence: 99%

“…While the technique is only applicable to chordal SDPs with bounded treewidths, it is able to reduce the cost of a size-n SDP all the way down to the cost of a size-n linear program, sometimes as low as O(τ 3 n). Indeed, chordal sparsity can be guaranteed in many important applications [32], [80], and software exist to automate the chordal reformulation [183].…”

Section: E Other Specialized Algorithmsmentioning

confidence: 99%

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Zhang

Josz

Sojoudi

2018

2018 Annual American Control Conference (ACC)

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Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.

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“…Numerical rounding on the eigenvalues of X k has already been used to reduce of rank(X), but penalizing rank(X k ) in optimization was not considered [12]. Minimum rank completions over linear matrix inequalities with general graphs has been performed in the context of optimal power flow, but few details were mentioned about how to penalize the rank of tree components [13].…”

Section: Introductionmentioning

confidence: 99%

Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

Miller

Zheng

Roig-Solvas

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. This paper leverages a minimum rank completion result to decompose the rank constraint on a single large matrix into multiple rank constraints on a set of smaller matrices. The re-weighted heuristic is used as a proxy for rank, and the specific form of the heuristic preserves the sparsity pattern between iterations. Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed. The problem of subspace clustering is used to demonstrate the computational improvement of the proposed method. 1 J. Miller, B. Roig-Solvas and M. Sznaier are with the ECE

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Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization

Cited by 31 publications

References 53 publications

Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

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