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Error bounds for monotone linear complementarity problems.(English)Zbl 0613.90095

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 complementarity 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.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C25 Convex programming

Cite

References:

[1]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).
[2]R.W. Cottle and G.B. Dantzig, ”Complementary pivot theory in mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125. ·Zbl 0155.28403 ·doi:10.1016/0024-3795(68)90052-9
[3]A.S. Householder,The Theory of Matrices in Numerical Analysis (Blaisdell Publishing, New York, 1964). ·Zbl 0161.12101
[4]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. ·Zbl 0505.90042
[5]O.L. Mangasarian, ”Simple computable bounds for solutions of linear complementarity problems and linear programs,”Mathematical Programming Study 25 (1985) 1–12. ·Zbl 0582.90098
[6]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. ·Zbl 0613.90066
[7]J.M. Ortega,Numerical Analysis: A Second Course (Academic Press, New York, 1972). ·Zbl 0248.65001
[8]J.-S. Pang, ”Inexact Newton methods for the nonlinear complementarity problem,” Report, School of Management, University of Texas at Dallas (Richardson, Texas, April 1985).
[9]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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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