Abstract
The minimizing problem for a convex program has a dual problem, that is, the maximizing problem of the Lagrangian. Although these problems have a duality gap in general, the duality gap can be eliminated by relaxing the constraint of the minimizing problem, so that the constraint is enforced only in the limit. We extend this result to the convexlike case. Moreover, we obtain a necessary condition for optimality for minimizing problems whose objective function and constraint mapping have convex Gateaux derivative.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Duffin, R. J.,Convex Analysis Treated by Linear Programming, Mathematical Programming, Vol. 4, No. 2, 1973.
Duffin, R. J.,Lagrange Multiplier Method for Convex Programs, Proceedings of the National Academy of Science of the USA, Vol. 72, No. 5, 1975.
Borwein, J. M.,A Note on Perfect Duality and Limiting Lagrangians, Mathematical Programming, vol. 18, No. 3, 1980.
Elster, K. H., andNehse, R.,Optimality Conditions for Some Nonconvex Problems, Springer-Verlag, New York, New York, 1980.
Fan, K.,Minimax Theorems, Proceedings of the National Academy of Sciences of the USA, Vol. 39, No. 1, 1953.
Nagahisa, Y., andSakawa, Y.,Nonlinear Programming in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 4, No. 3, 1969.
Rockafellar, R. T.,Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
Kelly, J. L., andNamioka, I.,Linear Topological Spaces, Springer-Verlag, New York, New York, 1963.
Author information
Authors and Affiliations
Additional information
Communicated by R. A. Tapia
The authors are indebted to Professor W. Takahashi of Tokyo Institute of Technology for his valuable comments, and also to the referees for their helpful suggestions.
Rights and permissions
About this article
Cite this article
Hayashi, M., Komiya, H. Perfect duality for convexlike programs. J Optim Theory Appl 38, 179–189 (1982). https://doi.org/10.1007/BF00934081
Issue Date:
DOI: https://doi.org/10.1007/BF00934081