Abstract
The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Fréchet differentiability of all operators involved, and the existence of the topological complement of the null space of the Fréchet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zangwill, W. T.,Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.
Evans, J. P., andGould, F. J.,Stability in Nonlinear Programming, Operations Research, Vol. 18, pp. 107–118, 1970.
Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.
Fiacco, A. V.,Sensitivity Analysis for Nonlinear Programming Using Penalty Methods, Mathematical Programming, Vol. 10, pp. 287–311, 1976.
Schechter, M.,Duality in Continuous Linear Programming, Journal of Mathematical Analysis and Applications, Vol. 37, pp. 139–141, 1972.
Kantorovich, L. V., andAkilov, G. P.,Functional Analysis in Normed Spaces, Translated from Russian by D. E. Brown, Pergamon Press, Oxford, England, 1964.
Nashed, M. Z., andVotruba, G. F.,A Unified Operator Theory of Generalized Inverses, Generalized Inverses and Applications, Edited by M. Z. Nashed, Academic Press, London, England, 1976.
Nashed, M. Z., andRall, L. B.,Annotated Bibliography on Generalized Inverses and Applications, Generalized Inverses and Applications, Edited by M. Z. Nashed, Academic Press, London, England, 1976.
Kadets, M. I., andMityagin, B. S.,Complemented Subspaces in Banach Spaces, Russian Mathematical Surveys, Vol. 28, pp. 77–95, 1973.
Buie, R. N., andAbrham, J.,Kuhn-Tucker Conditions and Duality in Continuous Programming, Utilitas Mathematica, Vol. 16, pp. 15–37, 1979.
Buie, R. M.,Continuous Programming, University of Toronto, Toronto, Ontario, Canada, Doctoral Dissertation, 1979.
Neustadt, L. W.,Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, New Jersey, 1976.
Hurwicz, L.,Programming in Linear Spaces, Studies in Linear and Nonlinear Programming, Edited by K. J. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, Stanford, California, 1958.
Ekeland, I., andTemam, R.,Analyse Convexe et Problemes Variationnels, Gauthier-Villars, Paris, France, 1973.
Taylor, A. E.,Introduction to Functional Analysis, Wiley, New York, New York, 1958.
Rudin, W.,Functional Analysis, McGraw-Hill Book Company, New York, New York, 1973.
Malik, H. J.,A Note on Generalized Inverses, Naval Research Logistics Quarterly, Vol. 15, pp. 605–612, 1968.
Caradus, S. R.,Operator Theory of the Pseudo-Inverse, Report No. 38, Queen's Papers in Pure and Applied Mathematics, Queen's University, Kingston, Ontario, Canda, 1974.
Groetsch, C. W.,Generalized Inverses of Linear Operators: Representation and Approximation, Marcel Dekker, New York, New York, 1977.
Dunford, N., andSchwartz, J. T.,Linear Operators, Part 1: General Theory, Interscience Publishers (John Wiley and Sons), New York, New York, 1966.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
The authors are indebted to the Natural Sciences and Engineering Research Council of Canada for financial support through Grants A4109 and A7329, respectively. They would also like to thank the referee for his comments.
Rights and permissions
About this article
Cite this article
Buie, R.N., Abrham, J. Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems. J Optim Theory Appl 45, 347–373 (1985). https://doi.org/10.1007/BF00938440
Issue Date:
DOI: https://doi.org/10.1007/BF00938440