Abstract
It is shown that each feasible point of a positive semidefinite linear complementarity problem which is not a solution of the problem provides a simple numerical bound for some or all components of all solution vectors. Consequently each pair of primal-dual feasible points of a linear program which are not optimal provides a simple numerical bound for some or all components of all primal-dual solution vectors. In addition we show that the existence of such numerical bounds is not only sufficient but is also necessary for the boundedness of solution vector components for both the linear complementarity problem and the dual linear programs.
Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grant MCS-8200632.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
I. Adler and D. Gale, “On the solutions of the positive semidefinite complementarity problem”, Technical Report ORC 75-12, Operations Research Center, University of California (Berkeley, 1975).
R.W. Cottle and G.B., Dantzig, “Complementary pivot theory of mathematical programming”, Linear Algebra and Its Applications 1 (1968) 103–125.
R.W. Cottle, F. Giannessi and J.-L. Lions, eds., Variational inequalities and complementarity problems (Wiley, New York, 1980).
R. Doverspike, “Some perturbation results for the linear complementarity problem”, Mathematical Programming 23 (1982) 181–192.
O.L. Mangasarian, Nonlinear programming (McGraw-Hill, New York, 1969).
O.L. Mangasarian, “A condition number for linear inequalities and linear programs”, in: G. Bamberg and O. Opitz, eds, Methods of Operations Research 43, Proceedings of the 6. Symposium über Operations Research, Universität Augsburg, September 7–9, 1981 (Verlagsgruppe Athenäum/Hain/Scriptor/Hanstein, Königstein, 1981), pp. 3–15.
O.L. Mangasarian, “Characterization of bounded solutions of linear complementarity problems”, Mathematical Programming Study 19 (1982) 153–166.
S.M. Robinson, “Bounds for error in the solution of a perturbed linear program”, Linear Algebra and Its Applications 6 (1973) 69–81.
S.M. Robinson, “Generalized equations and their solutions, Part I: Basic theory”, Mathematical Programming Study 10 (1979) 128–141.
A.C. Williams, “Marginal values in linear programming” Journal of the Society for Industrial and Applied Mathematics 11 (1963) 82–94.
A.C. Williams: “Boundedness relations for linear constraint sets”, Linear Algebra-and Its Applications 3 (1970) 129–141.
A.C. Williams, “Complementarity theorems for linear programming”, SIAM Review 12 (1970) 135–137.
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Professor Geogre B. Dantzig on the occasion of his 70th birthday.
Rights and permissions
Copyright information
© 1985 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
Mangasarian, O.L. (1985). Simple computable bounds for solutions of linear complementarity problems and linear programs. In: Cottle, R.W. (eds) Mathematical Programming Essays in Honor of George B. Dantzig Part II. Mathematical Programming Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121071
Download citation
DOI: https://doi.org/10.1007/BFb0121071
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00920-4
Online ISBN: 978-3-642-00921-1
eBook Packages: Springer Book Archive