A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

Burer, Samuel; Monteiro, Renato D. C.

doi:10.1007/s10107-002-0352-8

Cited by 692 publications

(816 citation statements)

References 19 publications

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“…To give a better appreciation of this phenomenon to the reader, we conducted the following experiment. We chose two state-of-the-art SDP solvers, SDPLR [7] and SDPA [25], and solved the SDP relaxation MIQCP-SDP of these box QP instances using them. Tables 20 and 21 report the computational results, while Table 6 reports the same in summarized form.…”

Section: Relaxation Valuesmentioning

confidence: 99%

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

2010

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A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − xx T 0 and disjunctive programming. On the other hand, their main drawback is their huge size, even for problems of moderate size. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on the box QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 sec even for larger instances with 100 variables; the SDP+RLT relaxations of the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.

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Section: Relaxation Valuesmentioning

confidence: 99%

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

2010

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“…. , m. Let F denote the feasible set of (1). The basic SDP relaxation of (1) is minimize A 0 • X + a T 0 x subject to A i • X + a T i x + α i ≤ 0 (i = 1, .…”

Section: The Problem and Existing Techniquesmentioning

confidence: 99%

“…Semidefinite programming (SDP) relaxations can provide tight bounds [6,11,12], but they can also be expensive to solve by classical interior-point methods [14]. Many researchers have thus suggested various alternatives to interior-point methods for improving solution times [1,2,4,8,15,19]. These efforts have been quite successful on many classes of SDP relaxations.…”

Section: Introductionmentioning

confidence: 99%

Faster, but weaker, relaxations for quadratically constrained quadratic programs

2013

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We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed blockdiagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.

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“…A similar idea has been used also in relaxation for the max cut problem [7,10]. Note that Júdice and coworkers consider a different generalized Rayleigh quotient function in their study [12] of the Eigenvalue Complementarity Problem (EiCP).…”

Section: Optimality Conditionsmentioning

confidence: 99%

Unconstrained formulation of standard quadratic optimization problems

Bomze

Grippo

Palagi

2010

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A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum-Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. We propose unconstrained reformulations of StQPs, by using different approaches. We test our method on clique problems from the DIMACS challenge.

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A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

Cited by 692 publications

References 19 publications

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

Faster, but weaker, relaxations for quadratically constrained quadratic programs

Unconstrained formulation of standard quadratic optimization problems

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