A Unified Theorem on SDP Rank Reduction

So, Anthony Man–Cho; Ye, Yinyu; Zhang, Jiawei

doi:10.1287/moor.1080.0326

Cited by 54 publications

(35 citation statements)

References 15 publications

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“…In other words, the average performance can be much better than the stated worst-case bounds for randomly generated instances. Recently, So et al [11] developed methods for finding approximate low rank solutions for linear matrix inequalities. Their results unify the approximation bounds of Nemirovski et al [9] and Luo et al [8] as special cases (rank being 1), when all the data matrices are positive semidefinite.…”

Section: Introductionmentioning

confidence: 99%

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

He¹,

Luo²,

Nie³

et al. 2008

SIAM J. Optim.

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In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{xIf one of A k 's is indefinite while others and C are positive semidefinite, we prove that the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2 ) when F is the real line R, and by O(m) when F is the complex plane C. This result is an extension of the recent work of Luo et al. [8]. For (2), we show that the same ratio is bounded from below by O(1/ log m) for both the real and complex case, whenever all but one of A k 's are positive semidefinite while C can be indefinite. This result improves the so-called approximate S-Lemma of Ben-Tal et al. [2]. We also consider (2) with multiple indefinite quadratic constraints and derive a general bound in terms of the problem data and the SDP solution. Throughout the paper, we present examples showing that all of our results are essentially tight.

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Section: Introductionmentioning

confidence: 99%

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

He¹,

Luo²,

Nie³

et al. 2008

SIAM J. Optim.

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show abstract

“…So solving Problem 2 amounts to finding a sparse solution to this SDP. Here "sparse" means that there are few non-zero entries in the solution y; this differs from other notions of "low-complexity" SDP solutions, such as the low-rank solutions studied by So, Ye and Zhang [36].…”

Section: Solving Problem 2 By Mmwummentioning

confidence: 94%

Sparse Sums of Positive Semidefinite Matrices

Silva

Harvey

Sato

2015

ACM Trans. Algorithms

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Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank.We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale (2007). We also highlight an interesting connection between these two algorithms.Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs.

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“…Through a simple randomized polynomial-time procedure proposed by AMC So et al [14], we can extract a feasible solution of (QTTP) from the optimal solution of its SDP relaxation. And the approximation ratio is Ω( 1 log m ) or Ω( 1 √ log m ) which depends on the magnitude of n 2 .…”

Section: Approximation Algorithms For the Homogeneous Qttp Problemmentioning

confidence: 99%

Quadratic third-order tensor optimization problem with quadratic constraints

Yang

Zhao

2014

Stat., optim. inf. comput.

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Quadratically constrained quadratic programs (QQPs) problems play an important modeling role in many diverse problems. These problems are in general NP hard and numerically intractable. Semidefinite programming (SDP) relaxations often provide good approximate solutions to these hard problems. For several special cases of QQP, e.g., convex programs and trust region subproblems, SDP relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper, we consider a certain QQP where the variable is neither vector nor matrix but a third-order tensor. This problem can be viewed as a generalization of the ordinary QQP with vector or matrix as it's variable. Under some mild conditions, we first show that SDP relaxation provides exact optimal solutions for the original problem. Then we focus on two classes of homogeneous quadratic tensor programming problems which have no requirements on the constraints number. For one, we provide an easily implemental polynomial time algorithm to approximately solve the problem and discuss the approximation ratio. For the other, we show there is no gap between the SDP relaxation and itself.Keywords quadratic third-order tensor programming, semidefinite relaxation, SDP relaxation, approximate algorithm MSC (2000): 15A69, 90C20, 90C26, 90C59

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A Unified Theorem on SDP Rank Reduction

Cited by 54 publications

References 15 publications

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

Sparse Sums of Positive Semidefinite Matrices

Quadratic third-order tensor optimization problem with quadratic constraints

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