Computational aspects of a branch and bound algorithm for quadratic zero-one programming

Pardalos, Pãnos M.; Rodgers, Gregory

doi:10.1007/bf02247879

Cited by 239 publications

(135 citation statements)

References 16 publications

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“…Pardalos and Rodgers [37], [38] solve (QP) by Branch-and-Bound using a preprocessing phase where they try to fix some of the variables. The test on fixing the variables is based on the gradient of (1), which reads 2Qy + c, and exploits the fact that if y * is the global solution of (QP), then y * is also optimal for the linear program min{(2Qy * + c) T y : y ∈ {0, 1} n }.…”

Section: Further Approachesmentioning

confidence: 99%

“…Pardalos and Rodgers ( [37]) have proposed a test problem generator for Unconstrained Quadratic Binary Programming. Their procedure generates a symmetric integer matrix Q to define the objective function for (QP), with the linear term c represented by the main diagonal of Q, and has several parameters to control the characteristics of the problem, namely:…”

Section: Instances Of (Qp)mentioning

confidence: 99%

“…If all the data are nonnegative, then the optimum is x = 0. If all the data are nonpositive then the optimum is x = e. Therefore, it makes sense (as is done by the Pardalos and Rodgers generator ( [37])) to generate instances where the data are randomly chosen with mean = 0.…”

Section: Comparing (Mc) and (Qp) Instancesmentioning

confidence: 99%

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Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

2008

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We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a "nearly optimal" solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process.We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem.The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. The Max-Cut ProblemThe Max-Cut problem is one of the basic NP-hard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g.,

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Section: Further Approachesmentioning

confidence: 99%

Section: Instances Of (Qp)mentioning

confidence: 99%

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Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

2008

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“…Among them are several exact methods using branch and bound or branch and cut (see, e.g., [8,22,23,42]), but the high computational complexity of UBQP and the breadth of applications it embraces has led to the finding that, apart from isolated cases, problems of sizes larger than n = 100 cannot be solved by these exact methods in a reasonable time.…”

Section: Introductionmentioning

confidence: 99%

A hybrid metaheuristic approach to solving the UBQP problem

Glover

Hao

2010

European Journal of Operational Research

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This paper presents a hybrid metaheuristic approach (HMA) for solving the Unconstrained Binary Quadratic Programming (UBQP) problem. By incorporating a tabu search procedure into the framework of evolutionary algorithms, the proposed approach exhibits several distinguishing features, including a diversification-based combination operator and a distance-and-quality based replacement criterion for pool updating. The proposed algorithm is able to easily obtain the best-known solutions for 31 large random instances up to 7000 variables (which no previous algorithm has done) and find new best solutions for 3 of 9 instances derived from the set partitioning problem, demonstrating the efficacy of our proposed algorithm in terms of both solution quality and computational efficiency. Furthermore, some key elements and properties of the HMA algorithm are also analyzed.

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“…Moreover, many combinatorial optimization problems pertaining to graphs such as determining maximum cliques, maximum cuts, maximum vertex packing, minimum coverings, maximum independent sets, maximum independent weighted sets are known to be capable of being formulated by the UBQP problem as documented in papers of Pardalos and Rodgers (1990), Pardalos and Xue (1994). A review of additional applications and formulations can be found in Kochenberger et al (2004Kochenberger et al ( , 2005, Alidaee et al (2008), Lewis et al (2008).…”

Section: Introductionmentioning

confidence: 99%

Diversification-driven tabu search for unconstrained binary quadratic problems

Glover

Hao

2010

4OR-Q J Oper Res

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This paper describes a Diversification-Driven Tabu Search (D 2 TS) algorithm for solving unconstrained binary quadratic problems. D 2 TS is distinguished by the introduction of a perturbation-based diversification strategy guided by long-term memory. The performance of the proposed algorithm is assessed on the largest instances from the ORLIB library (up to 2500 variables) as well as still larger instances from the literature (up to 7000 variables). The computational results show that D 2 TS is highly competitive in terms of both solution quality and computational efficiency relative to some of the best performing heuristics in the literature.

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Computational aspects of a branch and bound algorithm for quadratic zero-one programming

Cited by 239 publications

References 16 publications

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

A hybrid metaheuristic approach to solving the UBQP problem

Diversification-driven tabu search for unconstrained binary quadratic problems

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