Abstract
In the paper a method is presented which evaluates a solution of a linear programming problem. From the parameters of primal and dual program a symmetric matrix game is constructed. The search for the solution of this game is reduced to a problem of the constrained convex minimization, where the minimized function is the maximum of finite number of affine functions and a standard simplex is the feasible region and the minimal value of the function is known. The subgradient method of Polyak is applied to solve this problem. The algorithm converges geometrically. On each iteration the projection onto a standard simplex is applied, for which a combinatorial algorithm is presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cegielski, A., A geometrically convergent subgradient projection method in matrix games, submitted to Numerische Mathematik
Cegielski, A., An algorithm of the projection onto a standard simplex, submitted to Journal of Optimization Theory and Applications.
Karlin, S., Mathematical methods and theory in games, programming and economics, Vol. I, Reading, Massachusetts, Palo Alto, London, Addison-Wesley Publishing Company, Inc. 1962.
Michelot, C., A finite algorithm for finding the projection of a point onto the canonical simplex of ℝn, Journal of Optimization Theory and Applications, Vol. 50, No. 1, 195–200, (1986).
Polyak, B. T., Minimization of nonsmooth functionals, USSR Computational Mathematics and Math. Physics, 9, 14–39, (1969).
Szep, J., Forgo, F., Einfuehrung in die Spieltheorie, Budapest, Akademiai Kiado 1983.
Shor, N. Z., Minimization methods for non-differentiable functions, Berlin, Heidelberg, New York, Tokyo, Springer-Verlag 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Physica-Verlag Heidelberg
About this paper
Cite this paper
Cegielski, A. (1992). A Subgradient Projection Method in Linear Programming. In: Gritzmann, P., Hettich, R., Horst, R., Sachs, E. (eds) Operations Research ’91. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-48417-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-48417-9_20
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-0608-3
Online ISBN: 978-3-642-48417-9
eBook Packages: Springer Book Archive