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Duality theorems and an optimality condition for non-differentiable convex programming

Published online by Cambridge University Press:  09 April 2009

P. Kanniappan
Affiliation:
School of Mathematics, Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, India
Sundaram M. A. Sastry
Affiliation:
School of Mathematics, Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, India
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Abstract

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Necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Barbu, V. and Precupanu, Th., Convexity and optimization in Banach spaces (Sijthoff and Noordhoff, The Netherlands, 1978).Google Scholar
[2]Girsenov, I. V., Lectures on mathematical theory of extremum problems, (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
[3]Ioffe, A. D. and Tihomirov, V. M., Theory of extremal problems (Studies in Mathematics and its Applications, 6, North-Holland, Amsterdam, New York, Oxford, 1979).Google Scholar
[4]Kanniappan, P. and Sundaram, M. A. Sastry, ‘A duality theorem for non-differentiable convex programming with operatorial constraints’, Bull. Austral. Math. Soc. 22 (1980), 145152.CrossRefGoogle Scholar
[5]Kanniappan, P. and Sundaram, M. A. Sastry, ‘A subgradient converse duality theorem for a convex programming’, communicated to J. Math. Anal. Appl.Google Scholar
[6]Rockafellar, R. T., ‘Extension of Fenchel's duality theorem for convex functions,’ Duke Math. J. 33 (1966), 8189.CrossRefGoogle Scholar
[7]Schechter, M., ‘A subgradient duality theorem’, J. Math. Anal. Appl. 61 (1977), 850855.CrossRefGoogle Scholar
[8]Schechter, M., ‘More on subgradient duality’, J. Math. Anal. Appl. 71 (1979), 251262.CrossRefGoogle Scholar
[9]Wolfe, P., ‘A duality theorem for non-linear programming’, Quart. Appl. Math. 19 (1961), 239244.CrossRefGoogle Scholar