# Error bounds for nondegenerate monotone linear complementarity problems

- 135 Downloads
- 40 Citations

## Abstract

Error bounds and upper Lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution. The existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate. Thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem, the residual that bounds the distance from a solution point consists of the complementarity condition alone, whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it. This and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region {*z∣Mz + q ≥ 0, z* ≥ 0} with a single linear affine inequality constraint.

## Key words

Linear complementarity error bounds Lipschitz continuity## Preview

Unable to display preview. Download preview PDF.

## 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, CA, 1975).Google Scholar
- [2]R.W. Cottle and G.B. Dantzig, “Complementary pivot theory in mathematical programming,”
*Linear Algebra and its Applications*1 (1968) 103–125.Google Scholar - [3]M.C. Ferris, “Finite termination of the proximal point algorithm,” to appear in
*Mathematical Programming Series A.*Google Scholar - [4]A.J. Goldman and A.W. Tucker, “Theory of linear programming,” in: H.W. Kuhn and A.W. Tucker, eds.,
*Linear Inequalities and Related Systems*(Princeton University Press, Princeton, NY, 1956) pp. 53–97.Google Scholar - [5]A.S. Householder,
*The Theory of Matrices in Numerical Analysis*(Blaisdell, New York, 1964).Google Scholar - [6]O.L. Mangasarian, “Characterizations of bounded solutions of linear complementarity problems,”
*Mathematical Programming Study*19 (1982) 153–166.Google Scholar - [7]O.L. Mangasarian, “A simple characterization of solution sets of convex programs,”
*Operations Research Letters*7 (1988) 21–26.Google Scholar - [8]O.L. Mangasarian, “Least norm solution of non-monotone linear complementarity problems,” University of Wisconsin-Madison, Computer Sciences Technical Report #686 (Madison, WI, 1987); to appear in:
*Kantorovich Memorial Volume*(American Mathematical Society, Providence, RI).Google Scholar - [9]O.L. Mangasarian and R.R. Meyer, “Nonlinear perturbation of linear programs,”
*SIAM Journal on Control and Optimization*17 (1979) 745–752.Google Scholar - [10]O.L. Mangasarian and T.-H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems,”
*SIAM Journal on Control and Optimization*25 (1987) 583–595.Google Scholar - [11]O.L. Mangasarian and T.-H. Shiau, “Error bounds for monotone linear complementarity problems,”
*Mathematical Programming*36 (1986) 81–89.Google Scholar - [12]J.M. Ortega,
*Numerical Analysis: A Second Course*(Academic Press, New York, 1972).Google Scholar - [13]B.T. Polyak and N.V. Tretyakov, “Concerning an iterative method for linear programming and its economic interpretation,”
*Economics and Mathematical Methods*8(5) (1972) 740–751.Google Scholar - [14]R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”
*SIAM Journal on Control and Optimization*14 (1976) 877–898.Google Scholar - [15]A.W. Tucker, “Dual systems of homogeneous linear relations,” in: H.W. Kuhn and A.W. Tucker, eds.,
*Linear Inequalities and Related Systems*(Princeton University Press, Princeton, NY, 1956) pp. 3–18.Google Scholar