Abstract
We give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered. When the point satisfies the linear inequalities of the complementarity problem, the residual consists of the complementarity condition plus its square root. This latter term is essential and without it the error bound cannot hold. We also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Adler and D. Gale, “On the solutions of the positive semidefinite complementarity problem”, Report 75-12, Operations Research Center, University of California (Berkeley, August 1975).
R.W. Cottle and G.B. Dantzig, “Complementary pivot theory in mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125.
A.S. Householder,The Theory of Matrices in Numerical Analysis (Blaisdell Publishing, New York, 1964).
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 6.Symposium über Operations Research, Universität Augsburg, September 7–9, 1981 (Verlagsruppe Athenäum Hain/Scriptor/Hanstein, Konigstein 1981) pp. 3–15.
O.L. Mangasarian, “Simple computable bounds for solutions of linear complementarity problems and linear programs,”Mathematical Programming Study 25 (1985) 1–12.
O.L. Mangasarian and T.-H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,” Technical Report #599, Computer Sciences Department, University of Wisconsin, (Madison, June 1985) to appear inSIAM Journal on Control and Optimization.
J.M. Ortega,Numerical Analysis: A Second Course (Academic Press, New York, 1972).
J.-S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,” Report, School of Management, University of Texas at Dallas (Richardson, Texas, April 1985).
J.-S. Pang, “A posteriori error bounds for the linearly-constrained variational inequality problem,” Report, School of Management, University of Texas at Dallas, (Richardson, Texas, June 1985).
Author information
Authors and Affiliations
Additional information
Sponsored by the United States Army under contract No. DAAG29-80-C-0041. This material is based on research sponsored by National Foundation Grant DCR-8420963 and Air Force Office of Scientific Research Grant AFOSR-ISSA-85-00080.
Rights and permissions
About this article
Cite this article
Mangasarian, O.L., Shiau, T.H. Error bounds for monotone linear complementarity problems. Mathematical Programming 36, 81–89 (1986). https://doi.org/10.1007/BF02591991
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591991