A new bound for the quadratic assignment problem based on convex quadratic programming

Anstreicher, Kurt M.; Brixius, Nathan W.

doi:10.1007/pl00011402

Cited by 91 publications

(88 citation statements)

References 27 publications

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“…In [4], the authors incorporate a convex quadratic programming bound that was introduced by Anstreicher and Brixius in [3], into a branch and bound framework that was running on a computational grid. Their computations are considered to be among the most extensive computations ever performed to solve combinatorial optimization problems.…”

Section: The Quadratic Assignment Problemmentioning

confidence: 99%

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SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

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In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem. NotationThe space of p × q real matrices is denoted by R p×q , the space of k × k symmetric matrices is denoted by S k , and the space of k×k symmetric positive semidefinite matrices by S + k . We will sometimes also use the notation X 0 instead of X ∈ S + k , if the order of the matrix is clear from the context. For index sets α, β ⊂ {1, . . . , n}, we denote the submatrix that contains the rows of A indexed by α and the columns indexed by β as A(α, β). If α = β, the principal submatrix A(α, α) of A is abbreviated as A(α). The ith column of a matrix C is denoted by C :,i .We use I n to denote the identity matrix of order n, and e i to denote the ith standard basis vector. Similarly, J n and u n denote the n×n all-ones matrix and all-ones n-vector respectively, and 0 n×n is the zero matrix of order n. We will omit subscripts if the order is clear from the context. The set of n × n permutation matrices is denoted by Π n . We set E ij = e i e T j . The 'vec' operator stacks the columns of a matrix, while the 'diag' operator maps an n × n matrix to the n-vector given by its diagonal. The adjoint operator of 'diag' we denote by 'Diag'. The trace operator is denoted by 'tr'.The Kronecker product A ⊗ B of matrices A ∈ R p×q and B ∈ R r×s is defined as the pr × qs matrix composed of pq blocks of size r × s, with block ij given by a ij B, i = 1, . . . , p, j = 1, . . . , q.

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Section: The Quadratic Assignment Problemmentioning

confidence: 99%

“…In the sequel we describe the quadratic programming bound (QPB) that is introduced in [3]. Let V be an n × (n − 1) matrix whose columns form an orthonormal basis for the nullspace of u T n , and defineÂ := V T AV ,B := V T BV .…”

Section: The Quadratic Assignment Problemmentioning

confidence: 99%

SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

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“…A quadratic programming (QP) relaxation for the min-cut problem is derived in [14]. That convex QP relaxation is based on the QP relaxation for the QAP, see [15,25,26]. Numerical results in [14] show that QP bounds are weaker, but cheaper to compute than the strongest SDP bounds, see also Sect.…”

Section: Linear and Quadratic Programming Relaxationsmentioning

confidence: 99%

The min-cut and vertex separator problem

Rendl¹,

Sotirov

2017

Comput Optim Appl

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We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n + 1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size (n ≈ 300) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying nonconvex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.

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“…Recent developments have produced improved, that is, tighter, bounds. The new methodologies include the interior point bound by Resende et al (1995), the level-1 RLT-based dual-ascent bound by Hahn and Grant (1998), the dual-based bound by Karisch et al (1999), the convex quadratic programming bound by Anstreicher and Brixius (2001), the level-2 RLT interior point bound by Ramakrishnan et al (2002), the SDP bound by Roupin (2004), the lift-and-project SDP bound by Burer and Vandenbussche (2006), the bundle method bound by Rendl and Sotirov (2007), and the HahnHightower level-2 RLT-based dual-ascent bound by Adams et al (2007). The tightest bounds are the lift-and-project SDP bound and the two level-2 RLT-based bounds.…”

Section: Introductionmentioning

confidence: 99%

“…The tightest bounds are the lift-and-project SDP bound and the two level-2 RLT-based bounds. However, when taking speed and efficiency into consideration, the most competitive bounds are the level-1 RLTbased dual-ascent bound by Hahn and Grant (1998), the convex quadratic programming bound by Anstreicher and Brixius (2001), and the Hahn-Hightower level-2 RLT-based dual-ascent bound by Adams et al (2007).…”

Section: Introductionmentioning

confidence: 99%

A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

Hahn

Zhu²,

Guignard

et al. 2012

INFORMS Journal on Computing

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Abstract:We apply the level-3 Reformulation Linearization Technique (RLT3) to the Quadratic Assignment Problem (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm, based on this RLT3 formulation. This algorithm is not guaranteed to calculate the RLT3 lower bound exactly, but approximates it very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3-based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problems sizes larger than size 25 still presents a challenge due to the large memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP. NotationEntries of a matrix E of size mxnx…xp, indexed by i,j,…,k, are denoted e ij…k . Conversely, given numbers , one can form a corresponding matrix of appropriate size. Z(P) will denote the optimal value of optimization problem (P).

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A new bound for the quadratic assignment problem based on convex quadratic programming

Cited by 91 publications

References 27 publications

SDP Relaxations for Some Combinatorial Optimization Problems

SDP Relaxations for Some Combinatorial Optimization Problems

The min-cut and vertex separator problem

A Level-3 Reformulation-Linearization Technique-Based Bound for the Quadratic Assignment Problem

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