Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Structure of the efficient point set

Author: Đinh The Lục
Journal: Proc. Amer. Math. Soc. 95 (1985), 433-440
MSC: Primary 49A50; Secondary 90A14, 90C31
MathSciNet review: 806083
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ C$ be a nontrivial cone and $ X$ be a set in the $ n$-dimensional Euclidean space. Denote by $ {\text{E}}(X\vert C)$ the set of all efficient points of $ X$ with respect to $ C$. It will be proven that under some adequate assumptions $ {\text{E}}(X\vert C)$ is homeomorphic to a simplex while $ n = 2$, and for $ n > 2$ it is a contractible set. Furthermore, the set of all weak efficient points of $ X$ with respect to $ C$ is arcwise connected and its local contractibility is equivalent to being a retract of $ X$. The results presented in this study cover all topological properties of the efficient point set which have been obtained by Peleg and Morozov for the case when $ C$ is the nonnegative orthant.

References [Enhancements On Off] (What's this?)

  • [1] C. Berger, Topological space, Macmillan, New York, 1963.
  • [2] K. Bergstresser, A. Charnes and P. L. Yu, Generalization of domination structures and nondominated solutions in multicriteria decision making, J. Optim. Theory Appl. 18 (1976), 3-13. MR 0426810 (54:14743)
  • [3] K. Borsuk, Theory of retracts, PWN, Warsaw, 1967. MR 0216473 (35:7306)
  • [4] H. W. Corley, Duality theory for maximizations with respect to cones, J. Math. Anal. Appl. 84 (1981), 560-568. MR 639684 (83m:90077)
  • [5] G. Debreu, Valuation equilibrium and Pareto optimum, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 588-592. MR 0065896 (16:500i)
  • [6] J. Dugundji, Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Composito Math. 13 (1958), 229-246. MR 0113217 (22:4055)
  • [7] V. V. Morozov, On properties of the set of nondominated vectors, Vestnik Moskov. Univ. Comput. Sci. Cyber. 4 (1977), 51-61.
  • [8] B. Peleg, Topological properties of the efficient point set, Proc. Amer. Math. Soc. 35 (1972), 531-536. MR 0303916 (46:3052)
  • [9] V. V. Podinovskji and V. D. Nogin, Pareto optimal solutions in multicriteria optimization problems, Nauka, Moscow, 1982.
  • [10] S. Schecter, Structure of the demand function and Pareto optimal set with natural boundary conditions, J. Math. Econom. 5 (1978), 1-21. MR 497585 (80b:90011)
  • [11] S. Smale, Global analysis and economics. V: Pareto theory with constraints, J. Math. Econom. 1 (1976), 213-222. MR 0426826 (54:14759)
  • [12] A. R. Warburton, Quasiconcave vector maximization: Connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives, J. Optim. Theory Appl. 40 (1983), 537-557. MR 717176 (84k:90086)
  • [13] P. L. Yu and M. Zeleny, Set of all nondominated solutions in linear cases and a multicriteria simplex method, J. Math. Anal. Appl. 49 (1975), 430-468. MR 0421660 (54:9656)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 49A50, 90A14, 90C31

Retrieve articles in all journals with MSC: 49A50, 90A14, 90C31

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society