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Duality theorems for convex programming without constraint qualification

Published online by Cambridge University Press:  09 April 2009

P. Kanniappan
P. Kanniappan
Affiliation:
School of Mathematics Madurai Kamaraj UniversityMadurai-625 021 Tamil Nadu, India Department of Mathematics Gandhigram Rural InstituteGandhigram-624 302 Tamil Nadu, India
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Abstract

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Invoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

MSC classification

Secondary: 90C25: Convex programming 90C30: Nonlinear programming 90C48: Programming in abstract spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 253 - 266
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Barbu, V. and Precupanu, Th., Convexity and optimization in Banach space (Sijthoff and Noordhoff International Publishers, 1978).Google Scholar
[2]Ben-Israel, A., Ben-Tal, A. and Ziobec, S., ‘Optimality conditions in convex programming’, in Survey of mathematical programming, Proceedings of the IX International Symposium of Mathematical Programming, edited by Prekopa, A. (Hungarian Academy of Sciences, Budapest, Hungary and North-Holland, Amsterdam, 1978).Google Scholar
[3]Ben-Tal, A., Ben-Israel, A. and Zlobec, S., ‘Characterization of optimality in convex programming without a constraint qualification’, J. Optimization Theory Appl. 20 (1976), 417–437.CrossRefGoogle Scholar
[4]Borwein, J. M. and Wolkowicz, H., ‘Characterization of optimality for the abstract convex program with finite dimensional range’, J. Austral. Math. Soc. Ser. A 30 (1981), 390–411.CrossRefGoogle Scholar
[5]Craven, B. D. and Mond, B., ‘Sufficient Fritz John optimality conditions for non-differentiable convex programming’, J. Austral. Math. Soc. Ser. B 19 (1976), 462–468.CrossRefGoogle Scholar
[6]Holmes, R. B., A course on optimization and best approximation (Lecture Notes in Mathematics, 247, Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[7]Ioffe, A. D. and Tihomirov, V. M., Theory of extremal problems (North-Holland Publishing Company, Amsterdam, 1979).Google Scholar
[8]Kanniappan, P. and Sastry, Sundaram M. A., ‘A duality theorem for non-differentiable convex programming with operatorial constraints’, Bull. Austral. Math. Soc. 22 (1980), 145–152.CrossRefGoogle Scholar
[9]Kanniappan, P. and Sastry, Sundaram M. A., ‘Duality theorems and an optimality condition for non-differentiable convex programming’, J. Austral. Math. Soc. Ser. A 32 (1982).Google Scholar
[10]Kuhn, H. W. and Tucker, A. W., Non-linear programming, Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability, pages 481–492 (University of California Press, 1951).Google Scholar
[11]Mond, B., ‘A class of non-differentiable mathematical programming problems’, J. Math. Anal. Appl. 46 (1974), 169–174.CrossRefGoogle Scholar
[12]Mond, B. and Schechter, M., ‘A programming problem with an Lp norm in the objective function’, J. Austral. Math. Soc. Ser. B 19 (1976), 333–342.CrossRefGoogle Scholar
[13]Mond, B. and Zlobec, S., ‘Duality for non-differentiable convex programming, Utilitas Math. 15 (1974), 291–302.Google Scholar
[14]Peressini, A. L., Ordered topological vector spaces (Harper and Row, 1967).Google Scholar
[15], A. P. and Robertson, J. W., Topological vector spaces (Cambridge University Press, 1964).Google Scholar
[16]Rockafellar, R. T., Convex analysis (Princeton University Press, 1970).CrossRefGoogle Scholar
[17]Rockafellar, R. T., ‘Extension of Fenchel's duality theorem for a convex functions’, Duke Math. J. 33 (1966), 81–89.CrossRefGoogle Scholar
[18]Schechter, M., ‘A subgradient duality theorem’, J. Math. Anal. Appl. 61 (1977), 850–855.CrossRefGoogle Scholar
[19]Schechter, M., ‘More on subgradient duality’, J. Math. Anal. Appl. 71 (1979), 251–262.CrossRefGoogle Scholar
[20]Watson, G. A., ‘A class of programming problems whose objective function contains a norm’, J. Approximation Theorem 23 (1978), 401–411.CrossRefGoogle Scholar
[21]Wolfe, P., ‘A duality theorem for non-linear programming’, Quart. Appl. Math. 19 (1961), 239–244.CrossRefGoogle Scholar
[22]Zowe, J., ‘Subdifferentiability of convex functions with values in an ordered vector spaces’, Math. Scand. 34 (1974), 69–83.CrossRefGoogle Scholar