Mathematical Programming

, Volume 48, Issue 1–3, pp 437–445

# Error bounds for nondegenerate monotone linear complementarity problems

• O. L. Mangasarian
Article

## 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

## References

1. [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. [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. [3]
M.C. Ferris, “Finite termination of the proximal point algorithm,” to appear inMathematical Programming Series A. Google Scholar
4. [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. [5]
A.S. Householder,The Theory of Matrices in Numerical Analysis (Blaisdell, New York, 1964).Google Scholar
6. [6]
O.L. Mangasarian, “Characterizations of bounded solutions of linear complementarity problems,”Mathematical Programming Study 19 (1982) 153–166.Google Scholar
7. [7]
O.L. Mangasarian, “A simple characterization of solution sets of convex programs,”Operations Research Letters 7 (1988) 21–26.Google Scholar
8. [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. [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. [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. [11]
O.L. Mangasarian and T.-H. Shiau, “Error bounds for monotone linear complementarity problems,”Mathematical Programming 36 (1986) 81–89.Google Scholar
12. [12]
J.M. Ortega,Numerical Analysis: A Second Course (Academic Press, New York, 1972).Google Scholar
13. [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. [14]
R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.Google Scholar
15. [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