Using a Mixed Integer Programming Tool for Solving the 0–1 Quadratic Knapsack Problem

Billionnet, Alain; Soutif, Éric

doi:10.1287/ijoc.1030.0029

Cited by 41 publications

(23 citation statements)

References 8 publications

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“…In Chaovalitwongse, Pardalos and Prokopyev [16] (see also [36]), the feasible set is defined by linear and quadratic constraints in the binary variables. For the special case where the constraints are assignment constraints, compact formulations are proposed in Liberti [32] (see also the references therein), while Billionnet and Soutif [9] considered the 0-1 quadratic knapsack problem.…”

Section: Related Workmentioning

confidence: 99%

Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem

Hansen

Meyer

2009

Discrete Applied Mathematics

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We present and compare three new compact linearizations for the quadratic 0-1 minimization problem, two of which achieve the same lower bound as does the "standard linearization". Two of the linearizations require the same number of constraints with respect to Glover's one, while the last one requires n additional constraints where n is the number of variables in the quadratic 0-1 problem. All three linearizations require the same number of additional variables as does Glover's linearization. This is an improvement on the linearization of Adams, Forrester and Glover (2004) which requires n additional variables and 2n additional constraints to reach the same lower bound as does the standard linearization. Computational results show however that linearizations achieving a weaker lower bound at the root node have better global performances than stronger linearizations when solved by Cplex.

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Section: Related Workmentioning

confidence: 99%

Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem

Hansen

Meyer

2009

Discrete Applied Mathematics

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“…This heuristic was tested on twenty randomly-generated QKP instances, provided on-line by Billionnet and Soutif 1 , whose optimum solutions are known [2]. Ten of these instances consist of 100 objects and have density 25%; that is, 25% of their individual and quadratic values are non-zero.…”

Section: Tests On Random Instancesmentioning

confidence: 99%

Evolving heuristically difficult instances of combinatorial problems

Julstrom

2009

Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation

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When evaluating a heuristic for a combinatorial problem, randomly generated instances of the problem may not provide a thorough exploration of the heuristic's performance, and it may not be obvious what kinds of instances challenge or confound the heuristic. An evolutionary algorithm can search a space of problem instances for cases that are heuristically difficult. Evaluation in such an EA requires an exact algorithm for the problem, which limits the sizes of the instances that can be explored, but the EA's (small) results can reveal misleading patterns or structures that can be replicated in larger instances. As an example, a genetic algorithm searches for instances of the quadratic knapsack problem that are difficult for a straightforward greedy heuristic. The GA identifies such instances, which in turn reveal patterns that mislead the heuristic. c ACM, 2009. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in

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“…Billionnet and Soutif described exact algorithms for the quadratic (single-)knapsack problem [1] [2] and have posted on-line a collection of randomly-generated QKP instances 1 . From twenty of these, we have constructed 60 QMKP instances, with n = 100 and 200 objects and three to ten knapsacks.…”

Section: Some Qmkp Instancesmentioning

confidence: 99%

The quadratic multiple knapsack problem and three heuristic approaches to it

Hiley

Julstrom

2006

Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation

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The quadratic multiple knapsack problem extends the quadratic knapsack problem with K knapsacks, each with its own capacity C k . A greedy heuristic fills the knapsacks one at a time with objects whose contributions are likely to be large relative to their weights. A hill-climber and a genetic algorithm encode candidate solutions as strings over {0, 1, . . . , K} with length equal to the number of objects. The hill-climber's neighbor operator is also the GA's mutation. In tests on 60 problem instances, the GA performed better than the greedy heuristic on the smaller instances, but it fell behind as the numbers of objects and knapsacks grew. The hill-climber always outperformed the greedy heuristic, and on the larger instances, also the GA.

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Using a Mixed Integer Programming Tool for Solving the 0–1 Quadratic Knapsack Problem

Cited by 41 publications

References 8 publications

Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem

Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem

Evolving heuristically difficult instances of combinatorial problems

The quadratic multiple knapsack problem and three heuristic approaches to it

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