An exact method based on Lagrangian decomposition for the 0–1 quadratic knapsack problem
The 0–1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity constraint. In this paper we present an exact method to solve this problem. This method makes use of the computation of an upper bound for (QKP)...
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| Published in | European journal of operational research Vol. 157; no. 3; pp. 565 - 575 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
16.09.2004
Elsevier Elsevier Sequoia S.A |
| Series | European Journal of Operational Research |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0377-2217 1872-6860 |
| DOI | 10.1016/S0377-2217(03)00244-3 |
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| Abstract | The 0–1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity constraint. In this paper we present an exact method to solve this problem. This method makes use of the computation of an upper bound for (QKP) which is derived from Lagrangian decomposition. It allows us to find the optimum of instances with up to 150 variables whatever their density, and with up to 300 variables for medium and low density. |
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| AbstractList | The 0-1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity constraint. In this paper we present an exact method to solve this problem. This method makes use of the computation of an upper bound for (QKP) which is derived from Lagrangian decomposition. It allows us to find the optimum of instances with up to 150 variables whatever their density, and with up to 300 variables for medium and low density. [PUBLICATION ABSTRACT] The 0-1 quadratic knapsack problem (QKP) consists of maximizing a pseudo-Boolean quadratic function with positive coefficients subject to a linear capacity constraint. We present in this paper a new exact method for solving this problem. This method makes use of the computation of an upper bound for (QKP) using a technique derived from the Lagrangean decomposition methods. The method we use is applied to very large sized problems and allows to find the optimum of problems up to 150 variables whatever their density is and up to 300 variables for problems with medium and low density.KEY-WORDS:0-1 quadratic optimization, knapsack, Lagrangean decomposition, computational results, branch and bound. The 0–1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity constraint. In this paper we present an exact method to solve this problem. This method makes use of the computation of an upper bound for (QKP) which is derived from Lagrangian decomposition. It allows us to find the optimum of instances with up to 150 variables whatever their density, and with up to 300 variables for medium and low density. |
| Author | Soutif, Éric Billionnet, Alain |
| Author_xml | – sequence: 1 givenname: Alain surname: Billionnet fullname: Billionnet, Alain email: alain.billionnet@iie.cnam.fr organization: CEDRIC––IIE, 18 allée Jean Rostand, 91025 Evry Cedex, France – sequence: 2 givenname: Éric surname: Soutif fullname: Soutif, Éric email: eric.soutif@univ-paris1.fr organization: CERMSEM UMR CNRS 8095, Université Paris 1 Panthéon-Sorbonne, 106-112 boulevard de l'Hôpital, 75647 Paris Cedex 13, France |
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| References | Chardaire, Sutter (BIB5) 1995; 41 Billionnet, Faye, Soutif (BIB2) 1999; 112 Chaillou, Hansen, Mahieu (BIB4) 1986; Vol. 1403 Gallo, Hammer, Simeone (BIB6) 1980; 12 Hammer, Rader (BIB7) 1997; 35 Caprara, Pisinger, Toth (BIB3) 1999; 11 Billionnet, Calmels (BIB1) 1996; 92 Michelon, Veuilleux (BIB8) 1996; 92 Michelon (10.1016/S0377-2217(03)00244-3_BIB8) 1996; 92 Gallo (10.1016/S0377-2217(03)00244-3_BIB6) 1980; 12 Billionnet (10.1016/S0377-2217(03)00244-3_BIB1) 1996; 92 Chaillou (10.1016/S0377-2217(03)00244-3_BIB4) 1986; Vol. 1403 Billionnet (10.1016/S0377-2217(03)00244-3_BIB2) 1999; 112 Chardaire (10.1016/S0377-2217(03)00244-3_BIB5) 1995; 41 Hammer (10.1016/S0377-2217(03)00244-3_BIB7) 1997; 35 Caprara (10.1016/S0377-2217(03)00244-3_BIB3) 1999; 11 |
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| Snippet | The 0–1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity... The 0-1 quadratic knapsack problem (QKP) consists in maximizing a quadratic pseudo-Boolean function with positive coefficients subject to a linear capacity... The 0-1 quadratic knapsack problem (QKP) consists of maximizing a pseudo-Boolean quadratic function with positive coefficients subject to a linear capacity... |
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| SubjectTerms | 0–1 quadratic programming Branch-and-bound Computational Complexity Computational results Computer Science Knapsack Knapsack problem Lagrangian decomposition Mathematical programming Optimization Studies |
| Title | An exact method based on Lagrangian decomposition for the 0–1 quadratic knapsack problem |
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