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Duality without constraint qualification for multiobjective programming

Published online by Cambridge University Press:  17 February 2009

R. R. Egudo ,
T. Weir  and
B. Mond
R. R. Egudo
Affiliation:
School of App. Science, Monash University College, Churchill VIC. 3842.
T. Weir
Affiliation:
Department of Mathematics, Australian Defence Force Academy, Campbell, ACT 2600.
B. Mond
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, VIC 3083.
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Abstract

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Extending earlier duality results for multiobjective programs, this paper defines dual problems for convex and generalised convex multiobjective programs without requiring a constraint qualification. The duals provide multiobjective extensions of the classical duals of Wolfe and Schechter and some of the more recent duals of Mond and Weir.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 4 , April 1992 , pp. 531 - 544
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Ben-Israel, A., Ben-Tal, A. and Charnes, A., “Necessary and sufficient conditions for Pareto Optimum in convex programming”, Econometrica 45 (1977) 811–820.CrossRefGoogle Scholar
[2] Ben-Israel, A., Ben-Tal, A. and Zlobec, S., Optimality in nonlinear programming: a feasible directions approach, (Wiley-Interscience, New York, 1981).Google Scholar
[3] Ben-Israel, A. and Mond, B., “First order optimality conditions for generalized convex programming: a feasible directions approach”, Utilitas Mathematica 25 (1984) 249–262.Google Scholar
[4] Chew, K. L. and Choo, E. U., “Pseudolinearity and efficiency”, Mathematical Programming 28 (1982) 226–239.CrossRefGoogle Scholar
[5] Egudo, R. R., “Efficiency and generalized convex duality for multiobjective programs”, J. Math. Anal. Appl. 138 (1989) 84–94.CrossRefGoogle Scholar
[6] Egudo, R. R., “Multiobjective duality in mathematical programming”, PhD. Thesis, La Trobe University, Bundoora, Victoria, Australia, 1990.Google Scholar
[7] Geoffrion, A. M., “Proper efficiency and the Theory of Vector Maximization”, J. Math. Anal. Appl. 22 (1968) 618–630.CrossRefGoogle Scholar
[8] Karlin, S., Mathematical methods and theory in games, programming and economics, Vol 1, (Addison-Wesley, Reading Massachusetts, 1959).Google Scholar
[9] Kuhn, H. W. and Tucker, A. W., “Nonlinear programming”, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, (ed. Neyman, J.), (University of California Press, Berkeley, California, 1951), 481–493.Google Scholar
[10] Mangasarian, O. L., Nonlinear programming, (McGraw-Hill, New York, 1969).Google Scholar
[11] Mond, B. and Weir, T., “Generalized concavity and duality”, in Generalized concavity in optimization and economics, (eds. Schaible, S. and Ziemba, W. T.), (Academic Press, N.Y. 1981), 263–279.Google Scholar
[12] Mond, B. and Zlobec, S., “Duality for nondifferentiable programming without a constraint qualification”, Utilitas Mathematica 15 (1979) 291–302.Google Scholar
[13] Schechter, M., “A subgradient duality theorem”, J. Math. Anal. Appl. 61 (1977) 850–855.CrossRefGoogle Scholar
[14] Weir, T., “Proper efficiency and duality for vector valued optimization problems”, J Austral. Math. Soc. (Series A) 43 (1987) 21–35.CrossRefGoogle Scholar
[15] Weir, T., “On efficiency, proper efficiency and duality in multiobjective programming”, Asia Pacific J. of Operations Research 7 (1990) 46–54.Google Scholar
[16] Weir, T. and Mond, B., “Duality for generalized convex programming without a constraint qualification”, Utilitas Mathematica 31 (1987) 233–242.Google Scholar
[17] Weir, T. and Mond, B., “Multiple objective programming duality without a constraint qualification”, Utilitas Mathematica 39 (1991) 41–55.Google Scholar
[18] Wolfe, P., “A duality theorem for nonlinear programming”, Quart. Appl. Math. 19 (1961) 239–244.CrossRefGoogle Scholar
[19] Zlobec, S., “characterizations of Pareto Minima in convex multicriteria optimization” Apkace Matermatiky 29 (1984) 342–349.Google Scholar
[20] Zlobec, S., “optimization: III. Optimal realizations of multi-objective models” ptimization 17 (1986) 429–445.CrossRefGoogle Scholar