Abstract
Consider an extreme point (EP)x 0 of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox 0 is degenerate,x 0 is called a degenerate EP. Corresponding tox 0, there are several bases. We will characterize the set of all bases associated withx 0, denoted byB 0. The setB 0 can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB 0, there is a corresponding undirected graphG 0, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG 0 and cycling (e.g., in linear programs).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Karwan, M. H., Lofti, V., Telgen, J., andZionts, S.,Redundancy in Mathematical Programming, Springer Verlag, New York, New York, 1983.
Gal, T.,Determination of All Neighbors of a Degenerate Extreme Point in Polytopes, Fern Universität, Hagen, Discussion Paper No. 17b, 1978.
Adler, I., andDantzig, G. B.,Maximum Diameter of Abstract Polytopes, Mathematical Programming Study 1, pp. 20–40, 1974.
Balas, E., andPadberg, M. W.,Set Partitioning, Combinatorial Programming: Methods and Applications, Edited by B. Roy, D. Reidel Publishing Company, New York, New York, pp. 205–258, 1975.
Brucker, P.,Diskrete Parametrische Optimierungsprobleme und Wesentliche Effiziente Punkte, Zeitschrift für Operations Research, Vol. 16, pp. 189–197, 1972.
Korte, B.,Ganzzahlige Programmierung, Universität Bonn, Institut für Ökonometrie und Operations Research, Working Paper No. 7646-OR, 1976.
Roy, B., Editor,Combinatorial Programming: Methods and Applications, D. Reidel Publishing Company, New York, New York, 1975.
Gal, T.,Postoptimal Analyses, Parametric Programming, and Related Topics, McGraw-Hill, New York, New York, 1979.
Gal, T., andNedoma, J.,Multiparametric Linear Programming, Management Science, Vol. 19, pp. 406–422, 1972.
Yu, P., andZeleny, M.,Linear Multiparametric Programming by Multicriteria Simplex Method, Management Science, Vol. 23, pp. 159–170, 1976.
Ecker, J. G., Hegner, N. S., andKouada, I. A.,Generating all Maximal Efficient Faces for Multiple Objective Linear Programs, Journal of Optimization Theory and Applications, Vol. 39, pp. 353–381, 1980.
Gal, T.,A General Method for Determining the Set of All Solutions to a Linear Vector Maximum Problem, European Journal of Operational Research, Vol. 1, pp. 307–332, 1977.
Isermann, H.,The Enumeration of the Set of All Efficient Solutions for a Linear Multiple Objective Program, Operations Research Quarterly, Vol. 28, pp. 711–720, 1977.
Zeleny, M.,Linear Multiobjective Programming, Springer-Verlag, New York, New York, 1974.
Cunningham, W. H.,A Network Simplex Method, Mathematical Programming, Vol. 11, pp. 105–116, 1976.
Cunningham, W. H., andKlincewicz, J. G.,On Cycling in the Network Simplex Method, Mathematical Programming, Vol. 23, pp. 182–189, 1983.
Balinski, M. L.,An Algorithm for Finding All Vertices of Convex Polyhedral Sets, SIAM Journal, Vol. 9, pp. 72–88, 1961.
Manas, M., andNedoma, J.,Finding All Vertices of a Convex Polyhedron, Numerische Mathematik, Vol. 12, pp. 226–229, 1968.
Telgen, J.,Redundant and Nonbinding Constraints in Linear Programming Problems, Erasmus University, Rotterdam, Holland, Discussion Paper, 1977.
Charnes, A.,Optimality and Degeneracy in Linear Programming, Econometrica, Vol. 20, pp. 160–170, 1952.
Bland, R. G.,New Finite Pivoting Rules for the Simplex Method, Mathematics of Operations Research, Vol. 2, pp. 103–107, 1977.
Author information
Authors and Affiliations
Additional information
Communicated by P. L. Yu
Rights and permissions
About this article
Cite this article
Gal, T. On the structure of the set bases of a degenerate point. J Optim Theory Appl 45, 577–589 (1985). https://doi.org/10.1007/BF00939135
Issue Date:
DOI: https://doi.org/10.1007/BF00939135