Abstract
Integer programming problems with a concave cost function are often encountered in optimization models involving economics of scale. In this paper, we propose an efficient exact algorithm for solving concave knapsack problems. The algorithm consists of an iterative process between finding lower and upper bounds by linearly underestimating the objective function and performing domain cut and partition by exploring the special structure of the problem. The lower bound is improved iteratively via cutting and partitioning the domain. This iteration process converges to the optimality in a finite number of steps. Promising computational results are reported for large-scale concave knapsack problems with up to 1200 integer variables. Comparison results with other existing methods in the literature are also presented.
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References
R. Bellman S.E. Dreyfus (1962) Applied Dynamic Programming Princeton University Press Princeton NJ
H.P. Benson S.S. Erenguc (1990) ArticleTitleAn algorithm for concave integer minimization over a polyhedron Naval Research Logistics 37 515–525
H.P. Benson S.S. Erengue R. Horst (1990) ArticleTitleA note on adopting methods for continuous global optimization to the discrete case Annals of Operations Research 25 243–252
K.M. Bretthauer A.V. Cabot M.A. Venkataramanan (1994) ArticleTitleAn algorithm and new penalties for concave integer minimization over a polyhedron Naval Research Logistics 41 435–454
K.M. Bretthauer B. Shetty (1995) ArticleTitleThe nonlinear resource allocation problem Operations Research 43 670–683
K.M. Bretthauer B. Shetty (2002) ArticleTitleThe nonlinear knapsack problem–algorithms and applications European Journal of Operations Research 138 459–472 Occurrence Handle10.1016/S0377-2217(01)00179-5
K.M. Bretthauer B. Shetty (2002) ArticleTitleA pegging algorithm for the nonlinear resource allocation problem Computers and Operational Research 29 505–527 Occurrence Handle10.1016/S0305-0548(00)00089-7
A.V. Cabot S.S. Erengue (1986) ArticleTitleA branch and bound algorithm for solving a class of nonlinear integer programming problems Naval Research Logistics 33 559–567
M.W. Cooper (1981) ArticleTitleSurvey of methods of pure nonlinear integer programming Management Science 27 353–361
G.B. Dantzig (1957) ArticleTitleDiscrete variable extremum problems Operations Research 5 266–277
M. Djerdjour K. Mathur H.M. Salkin (1988) ArticleTitleA surrogate relaxation based on algorithm for a general class quadratic multi-dimensional knapsack problem Operations Research Letters 7 253–258 Occurrence Handle10.1016/0167-6377(88)90041-7
D. Hochbaum (1995) ArticleTitleA nonlinear knapsack problem Operation Research Letters 17 103–110 Occurrence Handle10.1016/0167-6377(95)00009-9
R. Horst N.V. Thoai (1998) ArticleTitleAn integer concave minimization approach for the minimum concave cost capacitated flow problem on networks OR Spektrum 20 47–53 Occurrence Handle10.1007/BF01545530
T. Ibaraki N. Katoh (1988) Resource Allocation Problems: Algorithmic Approaches MIT Press Cambridge MA
M.S. Kodialam H. Luss (1998) ArticleTitleAlgorithm for separable nonlinear resource allocation problems Operations Research 46 272–284
R.E. Marsten T.L. Morin (1978) ArticleTitleA hybrid approach to discrete mathematical programming Mathematical Programming 14 21–40 Occurrence Handle10.1007/BF01588949
S. Martello P. Toth (1990) Knapsack Problems: Algorithms and Computer Implementations Wiley New York
K. Mathur H.M. Salkin B.B. Mohanty (1986) ArticleTitleA note on a general non-linear knapsack problems Operations Research Letters 5 79–81 Occurrence Handle10.1016/0167-6377(86)90107-0
K. Mathur H.M. Salkin S. Morito (1983) ArticleTitleA branch and search algorithm for a class of nonlinear knapsack problems Operation Research Letters 2 55–60
T.L. Morin R.E. Marsten (1976) ArticleTitleAn algorithm for nonlinear knapsack problems Management Science 22 1147–1158
T.L. Morin R.E. Marsten (1976) ArticleTitleBranch and bound strategies for dynamic programming Operations Research 24 611–627
P.M. Pardalos N. Kovoor (1990) ArticleTitleAn algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds Mathematical Programming 46 321–328 Occurrence Handle10.1007/BF01585748
P.M. Pardalos J.B. Rosen (1988) ArticleTitleReduction of nonlinear integer separable programming problems’ International Journal of Computer Mathematics 24 55–64
J.B. Rosen P.M. Pardalos (1987) Constrained Global Optimization: Algorithms and Applications Springer-Verlag Berlin
X.L. Sun D. Li (2002) ArticleTitleOptimality condition and branch and bound algorithm for constrained redundancy optimization in series systems Optimization Engineering 3 53–65 Occurrence Handle10.1023/A:1016541912439
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*Research supported by the National Natural Science Foundation of China under Grants 79970107 and 10271073,and the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.
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Sun, X.L., Wang, F.L. & Li, D. Exact Algorithm for Concave Knapsack Problems: Linear Underestimation and Partition Method. J Glob Optim 33, 15–30 (2005). https://doi.org/10.1007/s10898-005-1678-6
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DOI: https://doi.org/10.1007/s10898-005-1678-6