Abstract
In 1967, Wets and Witzgall (Ref. 1) made, in passing, a connection between frames of polyhedral cones and redundancy in linear programming. The present work elaborates and formalizes the theoretical details needed to establish this relation. We study the properties of optimal value functions in order to derive the correspondence between problems in redundancy and the frame of a polyhedral cone. The insights obtained lead to schemes to improve the efficiency of procedures to detect redundancy in the areas of linear programming, stochastic programming, and computational geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wets, R. J. B., andWitzgall, C.,Algorithms for Frames and Lineality Spaces of Cones, Journal of Research of the National Bureau of Standards, Vol. 71B, pp. 1–7, 1967.
Tomlin, J. A., andWelch, J. S.,Finding Duplicate Rows in a Linear Programming Model, Operations Research Letters, Vol. 5, pp. 7–11, 1986.
Graham, R., andYao, F.,A Whirlwind Tour of Computational Geometry, The American Mathematical Monthly, Vol. 97, pp. 687–701, 1990.
Walkup, D. W., andWets, R. J. B.,Stochastic Programs with Recourse, SIAM Journal on Applied Mathematics, Vol. 15, pp. 1299–1314, 1965.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Birge, J. R., andWets, R. J. B.,On-Line Solution of Linear Programs Using Sublinear Functions, Technical Report No. 86-25, IOE Department, University of Michigan, Ann Arbor, Michigan, 1986.
Wets, R. J. B.,On Inf-Compact Mathematical Programs, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, Vol. 3, pp. 426–435, 1973.
Yang, K., andMurty, K. G.,New Iterative Methods for Linear Inequalities, Journal of Optimization Theory and Applications, Vol. 72, pp. 163–185, 1992.
Karwan, M. H., Lofti, V., Telgen, J., andZionts, S.,Redundancy in Mathematical Programming: A State-of-the-Art Survey, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, Vol. 206, 1983.
Berbee, H. C. P., Boender, C. G. E., Rinnooy Kan, A. H. G., Scheffer, C. L., Smith, R. L., andTelgen, J.,Hit-and-Run Algorithms for the Identification of Nonredundant Linear Inequalities, Mathematical Programming, Vol. 37, pp. 184–207, 1987.
Preparata, F. P., andShamos, M. I.,Computational Geometry: An Introduction, Springer-Verlag, New York, New York, 1985.
Gastwirth, J.,On Robust Procedures, Journal of the American Statistical Association, Vol. 65, pp. 929–948, 1966.
Wallace, S. W., andWets, R. J. B.,Preprocessing in Stochastic Programming: The Case of Linear Programs, ORSA Journal on Computing, Vol. 4, pp. 45–59, 1992.
Dula, J. H., Helgason, R. V., andHickman, B. L.,Preprocessing Schemes and a Solution Method for the Convex Hull Problem in Multidimensional Space, Computer Science and Operations Research: New Developments in Their Interfaces, Edited by O. Balci, Pergamon Press, Oxford, England, pp. 59–70, 1992.
Author information
Authors and Affiliations
Additional information
Communicated by E. Polak
The author is indebted to G. Dewan for his review and discussions.
Rights and permissions
About this article
Cite this article
Dulá, J.H. Geometry of optimal value functions with applications to redundancy in linear programming. J Optim Theory Appl 81, 35–52 (1994). https://doi.org/10.1007/BF02190312
Issue Date:
DOI: https://doi.org/10.1007/BF02190312