Abstract
We present sufficient and necessary conditions for classes of separable (additive) functions to generate the set of nondominated outcomes in multicriteria optimization problems. The basic technique consists of convexifying the set of outcomes and then applying the standard characterization of a convex set by a class of linear functions. The conditions include the case when the set of feasible alternatives is convex and the criteria are convex-transformable. We show that the sum of powers and the sum of functions of exponents can generate the nondominated set for an arbitrary set of outcomes (under compactness conditions). We also discuss monotonicity, proper nondominance, uniqueness and connectedness of solutions, and weights and trade-offs with respect to these functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.
Henig, M. I.,Proper Efficiency with Respect to Cones, Journal of Optimization and Applications, Vol. 36, pp. 387–407, 1982.
Benson, B.,An Improved Definition of Proper Efficiency for Vector Maximization with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 71, pp. 232–241, 1979.
Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contribution to the Theory of Games, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 87–91, 1953.
Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 613–630, 1968.
Yu, P. L.,A Class of Solutions for Group Decision Problems, Management Science, Vol. 19, pp. 936–946, 1973.
Zeleny, M.,Compromise Programming, Multiple Criteria Decision Making, Edited by J. L. Cochrane and M. Zeleny, University of South Carolina Press, Columbia, South Carolina, pp. 373–391, 1973.
Yu, P. L., andLeitmann, G.,Compromise Solution, Domination Structures, and Salukvadze's Solution, Journal of Optimization Theory and Applications, Vol. 13, pp. 362–378, 1974.
Salukvadze, M. E.,Optimization of Vector Functionals, Automatika i Telemenkhanika, pp. 5–15, 1971 (in Russian).
Huang, S. C.,Note on the Mean-Square Strategy of Vector-Valued Objective Function, Journal of Optimization Theory and Applications, Vol. 9, pp. 364–366, 1972.
Bowman, V. J.,On the Relationship of the Tchebycheff Norm and the Efficient Frontier of Multiple-Criteria Objectives, Multiple-Criteria Decision Making, Edited by H. Thiriez and S. Zionts, Springer-Verlag, New York, pp. 76–85, 1976.
Haimes, Y. Y., Lasdon, L., andWismer, D.,On a Bicriterion Formulation of the Problems of Integrated Systems Identification and System Optimization, IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-1, pp. 296–297, 1971.
Hocking, R. R., andShepard, R. L.,Parametric Solution of a Class of Nonconvex Program, Operations Research, Vol. 19, pp. 1742–1747, 1971.
Benson, H. P.,Vector Maximization with Two Objectives Functions, Journal of Optimization Theory and Applications, Vol. 28, pp. 253–257, 1979.
Wierzbicki, A. P.,A Mathematical Basis for Satisficing Decision Making, Organizations: Multiple Agents with Multiple Criteria, Edited by J. N. Morse, Springer-Verlag, New York, New York, pp. 465–485, 1981.
Gearhart, W. B.,Compromise Solutions and Estimation of the Noninferior Set, Journal of Optimization Theory and Applications, Vol. 28, pp. 29–47, 1979.
Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203–218, 1984.
Jahn, J.,Some Characterizations of the Optimal Solutions of a Vector Optimization Problem, OR Spektrum, Vol. 7, pp. 7–17, 1985.
Gearhart, W. B.,Characterization of Properly Efficient Solutions by Generalized Scalarization Methods, Journal of Optimization Theory and Applications, Vol. 41, pp. 491–502, 1983.
Avriel, M., andZang, I.,Generalized Convex Functions with Applications to Nonlinear Programming, Mathematical Programs for Activity Analysis, Edited by P. Van Moeseke, North-Holland, Amsterdam, The Netherlands, Chapter 2, 1974.
Hartley, R.,On Cone-Efficiency, Cone-Convexity, and Cone-Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.
Borwein, J.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.
Soland, R. M.,Multicriteria Optimization: A General Characterization of Efficient Solutions, Decision Sciences, Vol. 10, pp. 26–38, 1979.
Avriel, M.,Nonlinear Programming: Analysis and Methods, Prentice-Hall, Englewood Cliff, New Jersey, 1976.
Naccache, P. H.,Connectedness of the Set of Nondominated Outcomes in Multicriteria Optimization, Journal of Optimization Theory and Applications, Vol. 25, pp. 459–467, 1978.
Warburton, A. R.,Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537–557, 1983.
Chankong, V., andHaimes, Y. Y.,Multiobjective Decision Making: Theory and Methodology, North-Holland, New York, New York, 1983.
Author information
Authors and Affiliations
Additional information
Communicated by M. A. Avriel
The research was done while the author was Visiting Professor at the University of Illinois at Urbana-Champaign, Illinois. The author is grateful to G. Hazen, Northwestern University, for helpful discussions.
Rights and permissions
About this article
Cite this article
Henig, M.I. Characterizing the nondominated set by separable functions. J Optim Theory Appl 59, 423–444 (1988). https://doi.org/10.1007/BF00940308
Issue Date:
DOI: https://doi.org/10.1007/BF00940308