Abstract
In this paper, existence and characterization of solutions and duality aspects of infinite-dimensional convex programming problems are examined. Applications of the results to constrained approximation problems are considered. Various duality properties for constrained interpolation problems over convex sets are established under general regularity conditions. The regularity conditions are shown to hold for many constrained interpolation problems. Characterizations of local proximinal sets and the set of best approximations are also given in normed linear spaces.
Similar content being viewed by others
References
Chui, C. A., Deutsch, F., andWard, D.,Constrained Best Approximation in Hilbert Space, Constructive Approximation, Vol. 6, pp. 35–64, 1990.
Irvine, L. D., Marin, S. P., andSmith, P. W.,Constrained Interpolation and Smoothing, Constructive Approximation, Vol. 2, pp. 129–152, 1986.
Micchelli, C., andUtreras, F.,Smoothing and Interpolation in a Convex Subset of a Hilbert Space, SIAM Journal on Scientific and Statistical Computation, Vol. 9, pp. 728–746, 1988.
Ward, J.,Some Constrained Approximation Problems, Approximation Theory, Edited by C. Chui, L. Schumaker, and J. Ward, Academic Press, New York, New York, pp. 211–229, 1986.
Jeyakumar, V., andWolkowicz, H.,Zero Duality Gaps in Infinite-Dimensional Programming, Journal of Optimization Theory and Applications, Vol. 67, pp. 87–108, 1990.
Jeyakumar, V., andGwinner, J.,Inequality Systems and Optimization, Journal of Mathematical Analysis and Applications, Vol. 158, pp. 51–71, 1991.
Precupanu, T.,Closedness Conditions for the Optimality of a Family of Nonconvex Optimization Problems, Optimization, Vol. 15, pp. 339–346, 1984.
Rockafellar, R. T.,Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.
Jeyakumar, V.,Duality and Infinite-Dimensional Optimization, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 15, pp. 1111–1122, 1990.
Jeyakumar, V., andWolkowicz, H.,Generalizations of Slater's Constraint Qualifications for Infinite Convex Programs, Mathematical Programming, Vol. 57, pp. 85–101, 1992.
Borwein, J., andLewis, A.,Partially-Finite Convex Programming, Mathematical Programming, Vol. 57, pp. 15–48, 1992.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969.
Attouch, H., andBrezis, H.,Duality for the Sum of Convex Functions in General Banach Spaces, Aspects of Mathematics and Its Applications, Edited by J. Barroso, North-Holland, Amsterdam, Holland, pp. 125–133, 1986.
Robinson, S. M.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.
De Boor, C.,A Practical Guide to Splines, Springer-Verlag, Berlin, Germany, 1978.
Holmes, R. B.,A Course on Optimization and Approximation, Springer-Verlag, Berlin, Germany, 1972.
Dierieck, C.,Characterizations of Good and Best Approximations in an Arbitrary Subset of a Normed Linear Space, Approximation Theory, Edited by E. W. Cheney, Academic Press, New York, New York, Vol. 3, pp. 335–340, 1980.
Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, Wiley, New York, New York, 1984.
Mangasarian, O. L.,A Simple Characterization of Solution Sets of Convex Programs, Operations Research Letters, Vol. 7, pp. 21–26, 1988.
Burke, J. V., andFerris, M. C.,Characterization of Solution Sets of Convex Programs, Operations Research Letters, Vol. 10, pp. 57–60, 1991.
Author information
Authors and Affiliations
Additional information
Communicated by F. Giannessi
The author is grateful to the referee for helpful suggestions which have contributed to the final preparation of this paper. This research was partially supported by Grant A68930162 from the Australian Research Council.
Rights and permissions
About this article
Cite this article
Jeyakumar, V. Infinite-dimensional convex programming with applications to constrained approximation. J Optim Theory Appl 75, 569–586 (1992). https://doi.org/10.1007/BF00940493
Issue Date:
DOI: https://doi.org/10.1007/BF00940493