Abstract
Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.
Similar content being viewed by others
References
Gal, T.,Rim Multiparametric Linear Programming, Management Science, Vol. 21, pp. 567–575, 1975.
Orchard-Hays, W.,Advanced Linear Programming Computing Techniques, McGraw-Hill, New York, New York, 1968.
Willner, L. B.,On Parametric Linear Programming, SIAM Journal on Applied Mathematics, Vol. 15, pp. 1253–1257, 1967.
Courtillot, M.,On Varying All the Parameters in a Linear-Programming Problem and Sequential Solution of a Linear Programming Problem, Operations Research, Vol. 10, pp. 471–475, 1962.
Saaty, T. L.,Coefficient Perturbation of a Constrained Extremum, Operations Research, Vol. 7, pp. 294–302, 1959.
Williams, A. C.,Marginal Values in Linear Programming, SIAM Journal on Applied Mathematics, Vol. 11, pp. 82–94, 1963.
Mills, H. D.,Marginal Values of Matrix Games and Linear Programs, Linear Inequalities and Related Systems, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, 1956.
Koehler, G. J., andWhinston, A. B.,Sensitivity Analysis in Leontief Substitution Systems, Management Science, Vol. 26, pp. 1272–1280, 1980.
Martin, D. H.,On the Continuity of the Maximum in Parametric Linear Programming, Journal of Optimization Theory and Applications, Vol. 17, pp. 205–210, 1975.
Hogan, W. W.,Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591–603, 1973.
Dantzig, G. B., Folkman, J., andShapiro, N.,On the Continuity of the Minimum Set of a Continuous Function, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 519–548, 1967.
Bitran, G. R.,Linear Multiple Objective Problems with Interval Coefficients, Management Science, Vol. 26, pp. 694–706, 1980.
Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972.
Vangeldère, J.,Pareto Weak Minimality and Various Connected Properties, Proceedings of the Third Symposium on Operations Research, Vol. 31, Edited by W. Oettli and F. Steffens, University of Mannheim, Mannheim, West Germany, 1978.
Author information
Authors and Affiliations
Additional information
Communicated by A. V. Fiacco
This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.
The author would like to thank two anonymous reviewers for their most useful comments on earlier versions of this paper.
Rights and permissions
About this article
Cite this article
Benson, H.P. On the optimal value function for certain linear programs with unbounded optimal solution sets. J Optim Theory Appl 46, 55–66 (1985). https://doi.org/10.1007/BF00938759
Issue Date:
DOI: https://doi.org/10.1007/BF00938759