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A bisection-extreme point search algorithm for optimizing over the efficient set in the linear dependence case. (English) Zbl 0799.90101

The problem of maximizing a linear function on the set of efficient points of a linear vector maximization problem is investigated. It is assumed that the objective function of this problem is a linear combination of the objective functions on the linear vector maximization problem. A so-called bisection-extreme point search algorithm is presented for the solution of this problem. It is proved that this method finds a solution in a finite number of iterations.
Reviewer: J.Jahn (Erlangen)

MSC:

90C29 Multi-objective and goal programming
90C05 Linear programming
90-08 Computational methods for problems pertaining to operations research and mathematical programming
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[1] Bazaraa, M. S., Jarvis, J. J., and Sherali, H. D. (1990),Linear Programming and Network Flows, Wiley, New York. · Zbl 0722.90042
[2] Benson, H. P. (1986), An Algorithm for Optimizing over the Weakly-Efficient Set,European J. of Operational Research 25, 192-199. · Zbl 0594.90082 · doi:10.1016/0377-2217(86)90085-8
[3] Benson, H. P. (1991), An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set,J. of Global Optimization 1, 83-104. · Zbl 0739.90056 · doi:10.1007/BF00120667
[4] Benson, H. P. (1981), Finding an Initial Efficient Extreme Point for a Linear Multiple Objective Program,J. of the Operational Research Society 32, 495-498. · Zbl 0454.90075
[5] Benson, H. P. (1992), A Finite, Non-Adjacent Extreme Point Search Algorithm for Optimization over the Efficient Set,J. of Optimization Theory and Applications 73, 47-64. · Zbl 0794.90048 · doi:10.1007/BF00940077
[6] Benson, H. P. (1981), Optimization over the Efficient Set, Discussion Paper No.35, Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.
[7] Benson, H. P. (1984), Optimization over the Efficient Set,J. of Mathematical Analysis and Applications 98, 562-580. · Zbl 0534.90077 · doi:10.1016/0022-247X(84)90269-5
[8] Bolintineanu, S. (1990), Minimization of a Quasi-Concave Function over an Efficient Set, Mathematics Research Paper No. 90-15, Department of Mathematics, La Trobe University, Bundoora, Victoria, Australia.
[9] Dauer, J. P. (1991), Optimization over the Efficient Set Using an Active Constraint Approach,Zeitschrift für Operations Research 35, 185-195. · Zbl 0734.90081
[10] Dessouky, M. I., Ghiassi, M., and Davis, W. J. (1986), Estimates of the Minimum Nondominated Criterion Values in Multiple-Criteria Decision-Making,Engineering Costs and Production Economics 10, 95-104.
[11] Ecker, J. G. and Hegner, N. S. (1978), On Computing an Initial Efficient Extreme Point,J. of the Operational Research Society 29, 1005-1007. · Zbl 0385.90104
[12] Ecker, J. G. and Kouada, I. A. (1978), Finding All Efficient Extreme Points for Multiple Objective Linear Programs,Mathematical Programming 14, 249-261. · Zbl 0385.90105 · doi:10.1007/BF01588968
[13] Evans, G. W. (1984), An Overview of Techniques for Solving Multiobjective Mathematical Programs,Management Science 30, 1268-1282. · Zbl 0551.90090 · doi:10.1287/mnsc.30.11.1268
[14] Evans, J. P. and Steuer, R. E. (1973), Generating Efficient Extreme Points in Linear Multiple Objective Programming: Two Algorithms and Computing Experience, in J. L. Cochrane and M. Zeleny (eds.),Multiple Criteria Decision Making, University of South Carolina Press, Columbia, South Carolina, pp. 349-365.
[15] Forgo, F. (1988),Nonconvex Programming, Akademiai Kiado, Budapest. · Zbl 0241.90055
[16] Horst, R. and Tuy, H. (1990),Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin. · Zbl 0704.90057
[17] Isermann, H. (1977), The Enumeration of the Set of All Efficient Solutions for a Linear Multiple Objective Program,Operational Research Quarterly 28, 711-725. · Zbl 0372.90086 · doi:10.1057/jors.1977.147
[18] Isermann, H. and Steuer, R. E. (1987), Computational Experience Concerning Payoff Tables and Minimum Criterion Values over the Efficient Set,European J. of Operational Research 33, 91-97. · Zbl 0632.90074 · doi:10.1016/0377-2217(88)90257-3
[19] Martos, B. (1965), The Direct Power of Adjacent Vertex Programming Methods,Management Science 12, 241-252. · Zbl 0142.17002 · doi:10.1287/mnsc.12.3.241
[20] Murty, K. G. (1983),Linear Programming, Wiley, New York. · Zbl 0521.90071
[21] Pardalos, P. M. and Rosen, J. B. (1987),Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, Berlin. · Zbl 0638.90064
[22] Pardalos, P. M. and Rosen, J. B. (1986), Methods for Global Concave Minimization: A Bibliographic Survey,SIAM Review 28, 367-379. · Zbl 0602.90105 · doi:10.1137/1028106
[23] Philip, J. (1972), Algorithms for the Vector Maximization Problem,Mathematical Programming 2, 207-229. · Zbl 0288.90052 · doi:10.1007/BF01584543
[24] Reeves, G. R. and Reid, R. C. (1988), Minimum Values over the Efficient Set in Multiple Objective Decision Making,European J. of Operational Research 36, 334-338. · doi:10.1016/0377-2217(88)90125-7
[25] Rosenthal, R. E. (1985), Principles of Multiobjective Optimization,Decision Sciences 16, 133-152. · doi:10.1111/j.1540-5915.1985.tb01479.x
[26] Steuer, R. E. (1986),Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York. · Zbl 0663.90085
[27] Weistroffer, H. R. (1985), Careful Usage of Pessimistic Values Is Needed in Multiple Objectives Optimization,Operations Research Letters 4, 23-25. · Zbl 0569.90087 · doi:10.1016/0167-6377(85)90046-X
[28] Yu, P. L. (1985),Multiple-Criteria Decision Making, Plenum, New York. · Zbl 0643.90045
[29] Zeleny, M. (1982),Multiple Criteria Decision Making, McGraw-Hill, New York. · Zbl 0588.90019
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