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Duality for a non-differentiable programming problem

Published online by Cambridge University Press:  17 April 2009

J. Zhang  and
B. Mond
J. Zhang
Affiliation:
School of Mathematics, La Trobe University, Bundoora Vic 3083, Australia, e-mail: [email protected]
B. Mond
Affiliation:
School of Mathematics, La Trobe University, Bundoora Vic 3083, Australia, e-mail: [email protected]
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Abstract

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A generalised dual to a non-differentiable programming problem is given and duality established under general convexity and invexity conditions. A second order dual is also given and duality theorems proved under generalised second order invexity condition.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 29 - 44
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bector, C.R. and Bector, B.K., ‘(Generalized)-binvex functions and second order duality for a nonlinear programming problem’, Congr. Numer. 52 (1986), 37–52.Google Scholar
[2]Bector, C.R. and Chandra, S., ‘Second order duality with nondifferential functions’, (in preparation).Google Scholar
[3]Ben-Israel, A. and Mond, B., ‘What is invexity?’, J. Austral. Math. Soc. Ser. B 28 (1986), 1–9.CrossRefGoogle Scholar
[4]Chandra, S., Craven, B.D. and Mond, B., ‘Generalized concavity and duality with a square root term’, Optimization 16 (1985), 653–662.CrossRefGoogle Scholar
[5]Craven, B.D., ‘Duality for generalized convex fractional programs’, in Generalized concavity in optimization and economics, (Schaible, S. and Ziemba, W.T., Editors) (Academic Press, New York, 1981), pp. 473–489.Google Scholar
[6]Eisenberg, E., ‘Supports of a convex function’, Bull. Amer. Math. Soc. 68 (1962), 192–195.Google Scholar
[7]Hanson, M.A., ‘On sufficiency of the Kuhn-Tucker conditions’, J. Math. Anal. Appl. 80 (1981), 545–550.Google Scholar
[8]Mangasarian, O.L., ‘Second and higher-order duality in nonlinear programming’, J. Math. Anal. Appl. 51 (1975), 607–620.Google Scholar
[9]Mangasarian, O.L. and Fromovitz, S., ‘The Fritz John necessary optimality conditions in the presence of equality and inequality conditions’, J. Math. Anal. Appl. 17 (1967), 37–47.CrossRefGoogle Scholar
[10]Mond, B., ‘Second order duality for nonlinear programs’, Opsearch 11 (1974), 90–99.Google Scholar
[11]Mond, B., ‘A class of nondifferentiable mathematical programming problems’, J. Math. Anal. Appl. 46 (1974), 169–174.CrossRefGoogle Scholar
[12]Mond, B. and Schechter, M., ‘On a constraint qualification in a nondifferentiable programming problem’, Naval Res. Logist. Quart. 23 (1976), 611–613.CrossRefGoogle Scholar
[13]Mond, B. and Smart, I., ‘Duality with invexity for a class of nondifferentiable static and continuous programming problems’, J. Math. Anal. Appl. 141 (1989), 373–388.CrossRefGoogle Scholar
[14]Mond, B. and Weir, T., ‘Generalized concavity and duality’, in Generalized concavity in optimization and economics, (Schaible, S. and Ziemba, W. T., Editors) (Academic Press, New York, 1981), pp. 263–279.Google Scholar
[15]Mond, B. and Weir, T., ‘Generalized convexity and higher order duality’, J. Math. Sci. 16–18 (1981–1983), 74–94.Google Scholar
[16]Riesz, F. and Sz.-Nagy, B., Functional analysis, (translated from the 2nd French edition by Boron, Leo F.) (Frederick Ungar Publishing Co., New York, 1955).Google Scholar
[17]Wolfe, P., ‘A duality theorem for nonlinear programming’, Quart. Appl. Math. 19 (1961), 239–244.Google Scholar
[18]Wolkowitz, H., ‘An optimality condition for a nondifferentiable convex program’, Naval Res. Logist. Quart. 30 (1983), 415–418.Google Scholar