Abstract
The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.
Similar content being viewed by others
References
Al-Khayyal, F. A. and Falk, J. E. (1983), Jointly Constrained Biconvex Programming, Mathematics of Operations Research 8, 273–286.
Belenson, S. and Kapur, K. C. (1973), An Algorithm for Solving Multicriterion Linear Programming Problems with Examples, Operational Research Quarterly 24, 65–77.
Benayoun, R., de Montgolfier, J., Tergny, J., and Laritchev, O. (1971), Linear Programming with Multiple Objective Functions: Step Method (STEM), Mathematical Programming 1, 366–375.
Benson, H. P. (1986), An Algorithm for Optimizing over the Weakly-Efficient Set, European J. of Operational Research 25, 192–199.
Benson, H. P. (1978), Existence of Efficient Solutions for Vector Maximization Problems, J. of Optimization Theory and Applications 26, 569–580.
Benson, H. P. (1982), On the Convergence of Two Branch-and-Bound Algorithms for Nonconvex Programming Problems, J. of Optimization Theory and Applications 36, 129–134.
Benson, H. P. (1984), Optimization over the Efficient Set, J. of Mathematical Analysis and Applications 98, 562–580.
Benson, H. P. and Horst, R. (forthcoming), A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set, Computers and Mathematics with Applications.
Blankenship, J. W. and Falk, J. E. (1976), Infinitely Constrained Optimization Problems, J. of Optimization Theory and Applications 19, 261–281.
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.
Evans, G. W. (1984), An Overview of Techniques for Solving Multiobjective Mathematical Programs, Management Science 30, 1268–1282.
Falk, J. E. and Soland, R. M. (1969), An Algorithm for Separable Nonconvex Programming Problems, Management Science 15, 550–569.
Ghiassi, M., DeVor, R.E., Dessouky, M. I. and Kijowski, B. A. (1984), An Application of Multiple Criteria Decision Making Principles for Planning Machining Operations, IEEE Transactions 16, 106–114.
Hansen, P. (ed.) (1983), Essays and Surveys on Multiple Criteria Decision Making, Springer-Verlag, Berlin.
Hemming, T. (1978), Multiobjective Decision Making Under Certainty, Economic Research Institute of the Stockholm School of Economics, Stockholm.
Horst, R. (1976), An Algorithm for Nonconvex Programming Problems, Mathematical Programming 10, 312–321.
Horst, R. (1988), Deterministic Global Optimization with Partition Sets Whose Feasibility is Not Known. Application to Concave Minimization, Reverse Convex Constraints, D.C.-Programming and Lipschitzian Optimization, J. of Optimization Theory and Applications 58, 11–37.
Horst, R. (1990), Deterministic Global Optimization: Recent Advances and New Fields of Application, Naval Research Logistics (37, 433–471.
Horst, R. (1986), A General Class of Branch and Bound Methods in Global Optimization with Some New Approaches for Convave Minimization, J. of Optimization Theory and Applications 51, 271–291.
Horst, R., Thoai, N. V. and Benson, H. P. (forthcoming), Concave Minimization via Conical Partitions and Polyhedral Outer Approximation, Mathematical Programming.
Horst, R. and Tuy, H. (1990), Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin.
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.
Kok, M. and Lootsma, F. A. (1985), Pairwise-Comparison Methods in Multiple Objective Programming, with Applications in a Long-Term Energy-Planning Model, European J. of Operational Research 22, 44–55.
McCormick, G. P. (1976), Computability of Global Solutions to Factorable Nonconvex Programs: Part I-Convex Underestimating Problems, Mathematical Programming 10, 147–175.
Pardalos, P. M. and Rosen, J. B. (1982), Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, Berlin.
Pardalos, P. M. and Rosen, J. B. (1986), Methods for Global Concave Minimization: A Bibliographic Survey, SLAM Review 28, 367–379.
Philip, J. (1972), Algorithms for the Vector Maximization Problem, Mathematical Programming 2, 207–229.
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.
Rosenthal, R. E. (1985), Principles of Multiobjective Optimization, Decision Sciences 16, 133–152.
Steuer, R. E. (1986), Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York.
Thoai, N. V. and Tuy, H. (1980), Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research 5, 556–566.
Weistroffer, H. R. (1985), Careful Usage of Pessimistic Values is Needed in Multiple Objectives Optimization, Operations Research Letters 4, 23–25.
Yu, P. L. (1985), Multiple-Criteria Decision Making, Plenum, New York.
Zeleny, M. (1982), Multiple Criteria Decision Making, McGraw-Hill, New York
Author information
Authors and Affiliations
Additional information
Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.
Rights and permissions
About this article
Cite this article
Benson, H.P. An all-linear programming relaxation algorithm for optimizing over the efficient set. J Glob Optim 1, 83–104 (1991). https://doi.org/10.1007/BF00120667
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00120667