Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T10:49:58.536Z Has data issue: false hasContentIssue false

On converse duality for a nondifferentiable program

Published online by Cambridge University Press:  17 April 2009

T.R. Gulati
Affiliation:
Department of Mathematics, University of Roorkee, Roorkee 247672, India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A nonlinear nondifferentiable program with linear constraints is considered and a converse duality theorem is discussed. First we weaken an assumption previously made by Bhatia, and then give a simple proof under this weaker hypothesis, using the Fritz John conditions. Finally, defining a generalized Slater constraint qualification which implies Abadie's constraint qualification, we give a simple condition for the dual problem to satisfy this constraint qualification.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Abadie, J., “On the Kuhn-Tucker theorem”, Nonlinear programming, 1936 (North Holland, Amsterdam, 1967).Google Scholar
[2]Bazaraa, M.S., Shetty, C.M., Foundations of optimization (Lecture Notes in Economics and Mathematical Systems, 122. Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[3]Bhatia, Davinder, “A note on duality theorem for a nonlinear programming problem”, Management Sci. 16 (1969/1970), 604606.CrossRefGoogle Scholar
[4]Craven, B.D. and Mond, B., “On converse duality in nonlinear programming”, Oper. Res. 19 (1971), 10751078.CrossRefGoogle Scholar
[5]Eisenberg, E., “Duality in homogeneous programming”, Proc. Amer. Math. Soc. 12 (1961), 783787.CrossRefGoogle Scholar
[6]Eisenberg, E., “Supports of a convex function”, Bull. Amer. Math. Soc. 68 (1962), 192195.CrossRefGoogle Scholar
[7]Francis, Richard L. and Cabot, A. Victor, “Properties of a multi-facility location problem involving Euclidian distances”, Naval Res. Logist. Quart. 19 (1972), 335353.CrossRefGoogle Scholar
[8]Mangasarian, Olvi L., Nonlinear programming (McGraw-Hill, New York, London, Sydney, 1969).Google Scholar
[9]Mond, Bertram, “Nonlinear nondifferentiable programming in complex space”, Nonlinear programming, 385400 (Proc. Sympos. Mathematics Research Centre, University of Wisconsin, Madison, 1970. Academic Press, New York, London, 1970).CrossRefGoogle Scholar
[10]Mond, Bertram, “A class of nondifferentiable mathematical programming problems”, J. Math. Anal. Appl. 46 (1974), 169174.CrossRefGoogle Scholar
[11]Mond, Bertram and Schechter, Murray, “On a constraint qualification in a nondifferentiable programming problem”, Naval Res. Logist. Quart. 23 (1976), 611613.CrossRefGoogle Scholar
[12]Peterson, David W., “A review of constraint qualifications in finite-dimensional spaces”, SIAM Rev. 15 (1973), 639654.CrossRefGoogle Scholar
[13]Sinha, S.M., “A duality theorem for nonlinear programming”, Management Sci. 12 (1966), 385390.CrossRefGoogle Scholar