Abstract
In this paper, a new algorithm to solve a general 0–1 programming problem with linear objective function is developed. Computational experiences are carried out on problems where the constraints are inequalities on polynomials. The solution of the original problem is equivalent with the solution of a sequence of set packing problems with special constraint sets. The solution of these set packing problems is equivalent with the ordering of the binary vectors according to their objective function value. An algorithm is developed to generate this order in a dynamic way. The main tool of the algorithm is a tree which represents the desired order of the generated binary vectors. The method can be applied to the multi-knapsack type nonlinear 0–1 programming problem. Large problems of this type up to 500 variables have been solved.
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References
Balas, E. and Mazzola, J. B., (1984) Nonlinear 0–1 Programming: II. Dominance Relations and Algorithms,Mathematical Programming,30, 22–45.
Beresnev, V.L. (1979), Algorithms for the minimization of polynomials with Boolean variables (Russian),Problemy Kibernetiki (Moscow) 36, 225–246.
Boros, E. (1985), Private communication.
Granot, D., Granot, F. and Kallberg, J. (1979), Converting Relaxation for Positive 0–1 Polynomial Programs,Mng. Sci. 25, 264–273.
Hansen, P., Jaumard, B., and Mathon, V. (1989), Constrained Nonlinear 0–1 Programming, RUTCOR Research Report, RRR # 47–89, November 1989 (to appear inORSA Journal in Computing).
Kolesar, P. (1980), Testing for Vision Loss in Glaucoma Suspects,Management Science,26, 439–149.
Maga, F. and Vizvári, B. (1986), The Relaxation of a Special Polynomial Zero-One Programming Problem to set Covering Problem,Alkalmazott Matematikai Lapok 12 41–49.
Nicoloso, S. and Nobili, P. (1990), A Set Covering Formulation of the Matrix Equipartition Problem, Istituto di Analisi dei Sistemi ed Informatica, R.311, November 1990.
Nijenhaus, A. and Wilf, H.S. (1975),Combinatorial Algorithms, Academic Press, New York.
Pardalos, P.M. and Li, Y. (1993), Integer Programming, inHandbook of Statistics, Vol. 9. (Editor C.R. Rao), Elsevier, 279–302.
Pardalos, P.M., Phillips, A.T. and Rosen, J.B. (1993),Topics in Parallel Computing in Mathematical Programming, Science Press.
Schoch, M. and Lyska, W. (1978), Kombinatorische Algorithmen zur Lösung spezieller nichtlinearer 0–1 Optimierungsaufgaben,Mathematische Operationsforschung und Statistik, Ser. Optimization 9, 9–20.
Tarjan, R.E., Data Structures and Network Flows, CBMS-NSF, Regional conference series in applied mathematics, vol.44.
Vizvári, B. (1975), Enumerative Methods in Polynomial 0–1 Programming, (Hungarian),Alkalmazott Matematikai Lapok 1, 373–384.
Wang, X.D. (1988), An Algorithm for Nonlinear 0–1 Programming and Its Application in Structural Optimization,J. Num. Method & Comp. Appl. 9, 22–31.
Zak, Y.A. (1978), Algorithms for Nonlinear Pseudo-Boolean Programming,Engineering Cybernetics 16, 29–40.
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Vizvári, B., Yilmaz, F. An ordering (enumerative) algorithm for nonlinear 0-1 programming. J Glob Optim 5, 277–290 (1994). https://doi.org/10.1007/BF01096457
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DOI: https://doi.org/10.1007/BF01096457