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Semicontinuity of multifunctions connected with optimization with respect to cones

Part of: Maps and general types of spaces defined by maps Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

Alicja Sterna-Karwat
Alicja Sterna-Karwat
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
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Abstract

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This paper studies topological upper and lower semicontinuity of the minimal value multifunction and the solution multifunction for optimization problems, which are defined in terms of cones, subject to perturbations in constraints. It extends the results of Tanino and Sawaragi to finite dimensions and one of Berge to multiple objective optimization problems.

MSC classification

Secondary: 46A40: Ordered topological linear spaces, vector lattices 54C60: Set-valued maps 90C30: Nonlinear programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 2 , April 1986 , pp. 183 - 193
Copyright
Copyright Australian Mathematical Society 1986

References

1Bednarczuk, E., On upper semicontinuity of global minima in constrained optimization problems, J. Math. Anal. Appl. 86 (1982), 309318.CrossRefGoogle Scholar
2Berge, C., Topological spaces (Macmillan Company, New York, 1963).Google Scholar
3Borwein, J., Optimization with respect to partial orderings (Ph D Thesis, Oxford University, 1974).Google Scholar
4Borwein, J., Convex relations in analysis and optimization, pp. 335377 in Generalized concavity in optimization and economics, eds. Schaible, S. and Ziemba, W. T. (Academic Press, London, 1981).Google Scholar
5Cesari, L. and Suryanarayna, M. B., Existence theorems for Pareto optimization; Multivalued and Banach space valued functionals, Trans. Amer. Math. Soc. 224 (1978), 3765.CrossRefGoogle Scholar
6Corley, H. W., An existence result for minimizations with respect to cones, J. Optim. Theory Appl. 31 (1980), 277281.CrossRefGoogle Scholar
7Corley, H. W., Duality theory for maximization with respect to cones, J. Math. Anal. Appl. 84 (1981), 560568.CrossRefGoogle Scholar
8Craven, B. D., Strong vector minimization and duality, Z. Angew. Math. Phys. 60 (1980), 15.CrossRefGoogle Scholar
9Craven, B. D., Vector-valued optimization in generalized concavity, pp. 661687 in Optimization and economics, eds. Schaible, S. and Ziemba, W. T. (Academic Press, London, 1981).Google Scholar
10Delahaye, J. P. and Denel, J., The continuities of the point-to-set maps, definitions and equivalences, Math. Programming Stud. 10 (1979), 812.CrossRefGoogle Scholar
11Hogan, W., Point-to-set maps in mathematical programming, SIAM Rev. 15 (1973), 591603.CrossRefGoogle Scholar
12Kelly, J. L., Linear topological spaces (Van Nostrand, Princeton, N.J., 1963).CrossRefGoogle Scholar
13Kuratowski, K., Topology (Academic Press, New York and Polish Scientific Publishers, Warszawa, 1966).Google Scholar
14Lin, J. G., Maximal vectors and multi-objective optimization, J. Optim. Theory Appl. 18 (1976), 4164.CrossRefGoogle Scholar
15Peressini, A. L., Ordered topological vector spaces (Harper and Row, New York, Evanston, London, 1967).Google Scholar
16Rolewicz, S., On sufficient conditions of vector optimization, Methods of Oper. Res. 43 (1981), 151156.Google Scholar
17Tanino, T. and Sawaragi, Y., Stability of non dominated solutions in multicriteria decision-making, J. Optim. Theory. Appl. 30 (1980), 229253.CrossRefGoogle Scholar
18Wagner, D. H., Semi-compactness with respect to Euclidean cone, Canad. J. Math. 29 (1977), 2936.CrossRefGoogle Scholar