An exact method based on Lagrangian decomposition for the 0–1 quadratic knapsack problem

Billionnet, Alain; Soutif, Éric

doi:10.1016/s0377-2217(03)00244-3

Cited by 95 publications

(48 citation statements)

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“…If the computed upper bound is strictly lower than the value of the best known feasible solution then we can conclude that all the variables indexed in I can definitively be fixed to 0. A similar constraint can be added for fixing variables to 1 instead of 0 as explained in [1] (Billionnet and Soutif, 2004). …”

Section: Fixing Some 0-1 Variables To Their Optimal Valuementioning

confidence: 99%

“…The integer quadratic multidimensional knapsack problem (QMKP ), which is known to be NP-hard [13] (Lueker, 1975), involves the maximization of a concave quadratic and separable function over a convex set of linear constraints 1 A extended abstract of this paper appeared in LNCS 4362 pp. 456 …”

Section: Introductionmentioning

confidence: 99%

“…However, most of the knapsack problems literature has addressed attention to specialized versions. For instance, there are large bodies of literature focusing on knapsack problems with linear objective function and a single constraint or m contraints (see [6] (Fayard and Plateau, 1974), [14] (Martello and Toth, 1983), [7] ( Fréville and Hanafi, 2005), on knapsack problem with the quadratic objective function of (QMKP ) over a single linear constraint and 0-1 or pure integer variables (see [1] ( Billionnet and Soutif, 2004), [16] (Pisinger et al, 2007), [2], [15] (Mathur and Salkin, 1983)). Nevertheless several applications require the use of pure integer variables and m capacity constraints, such as capital budgeting.…”

Section: Introductionmentioning

confidence: 99%

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Exact solution method to solve large scale integer quadratic multidimensional knapsack problems

2007

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In this paper we develop a branch-and-bound algorithm for solving a particular integer quadratic multi-knapsack problem. The problem we study is defined as the maximization of a concave separable quadratic objective function over a convex set of linear constraints and bounded integer variables. Our exact solution method is based on the computation of an upper bound and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process. We lead a numerical comparison between our method and three other existing algorithms. The approach we propose outperforms other procedures for large-scaled instances (up to 2000 variables and constraints).

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Section: Fixing Some 0-1 Variables To Their Optimal Valuementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact solution method to solve large scale integer quadratic multidimensional knapsack problems

2007

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“…Classical exact methods [39] for solving the QKP are the branch and bound (B&B) algorithms, where numerous upper bounds have been obtained, by using techniques such as derivation of upper planes [20], Lagrangian relaxation [24], reformulation [10], linearization [3], Lagrangian decomposition [4,5,41], semidefinite relaxation [26], and reduction strategies [24,45], etc.…”

Section: Introductionmentioning

confidence: 99%

“…In Section IV, the implementation of a miniSwarm optimizer for the QKP is described. In Section V, the extensive experimental results by the mini-Swarm, using an online collection of the QKP instances, are compared with those of some existing algorithms [5,32]. Finally, this paper is concluded in the Section VI.…”

Section: Introductionmentioning

confidence: 99%

A Mini-Swarm for the Quadratic Knapsack Problem

Xie

Liu

2007

2007 IEEE Swarm Intelligence Symposium

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Abstract-The 0-1 quadratic knapsack problem (QKP) is a hard computational problem, which is a generalization of the knapsack problem (KP). In this paper, a mini-Swarm system is presented. Each agent, realized with minor declarative knowledge and simple behavioral rules, searches on a structural landscape of the problem through the guided generate-and-test behavior under the law of socially biased individual learning, and cooperates with others by indirect interactions. The formal decomposition of behaviors allows understanding and reusing elemental operators, while utilizes the heuristic information on the landscape. The results on a collection of the QKP instances by mini-Swarm versions are compared with that of both a branch-and-bound algorithm and a greedy genetic algorithm, which show its effectiveness. I. INTRO DUCTIONThe 0-1 quadratic knapsack problem (QKP) was introduced by Gallo et al. [20], which consists in choosing elements from n items for maximizing a quadratic profit objective function subject to a linear capacity constraint. The QKP is a generalization of the 0-1 knapsack problem (KP) [38], by restricting all the quadratic coefficients to zero. The knapsack problems [38] have been intensively studied due to both its theoretical interest and its wide practical applicability. Due to its generality, the QKP has more practically applications [3,20], moreover, several graph problems can be formulated as the QKP [4,45].The KP is NP-hard, although it can be solved exactly in pseudo-polynomial time through dynamic programming based on Bellman recursion [21]. As a generalized version, the QKP is much harder than the KP [10,45], where the graph structure associated with coefficients of objective function plays a important role in its solution [46].Due to the NP-hardness in strong sense, the research efforts on solving the QKP may focus on exact methods, approximate algorithms and heuristic methods.Classical The studies on approximate algorithms, which seek guaranteed polynomial-time performance on approximate solution, were only limited on some special cases of the QKP, such as the classical KP [30], and the QKP where the underlying graph is edge series-parallel [46], etc.The heuristic methods arise from practical interest, which try to find good solution in high probability with acceptable computational efforts by considering certain imprecise knowledge on the structure of the problem. The linearization and exchange (LEX) heuristic [24] uses the best linear L2-approximation to build an associated linear KP and use it for determining items in a greedy way to form a good initial solution, which is then improved by an exchange method between certain items. Hua et al. [29] studied on the convex QKP by an approximate dynamic programming approach. Julstrom [32] also illustrated the capability of several heuristics, including greedy, genetic and greedy genetic algorithms on the QKP.Swarm algorithms [7,16,34] address on tackling a specified optimization task by multiple interacting agents, where each agent works u...

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