Abstract
A number of equivalent characterizations for the existence and boundedness of solutions of the linear complementarity problem: Mx+q≥0, x≥0, x T(Mx+q)=0 where M is an n×n real matrix and q is an n-vector, are given for the case when M is copositive plus. The special case when M is skew-symmetric covers the linear programming case. One useful characterization of existence and boundedness of solutions is given by solving a simple linear program. Other important characterizations are the Slater constraint qualification and the stability condition that for all arbitrary but sufficiently small perturbations of the data M and q which maintain copositivity plus, the perturbed linear complementarity problem is solvable and its solutions are uniformly bounded. An interesting sufficient condition for boundedness of solutions is that the linear complementarity problem have a nondegenerate vertex solution. Another result is that the subclass ™ of copositive plus matrices for which the linear complementarity problem has a solution for each q in R n, that is ™⊂Q, coincides with the subclass of copositive plus matrices for which the linear complementarity problem has a nonempty bounded solution set for each q in R n.
Research supported by National Science Foundation Grant MCS-7901066.
Preview
Unable to display preview. Download preview PDF.
References
R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”, SIAM Journal on Applied Mathematics 14 (1966) 147–158.
R.W. Cottle, “Solution rays for a class of complementary problems”, Mathematical Programming Study 1 (1974) 59–70.
R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”, Linear Algebra and its Applications 1 (1968) 103–125.
R. Doverspike, “Some perturbation results for the linear complementarity problem”, Mathematical Programming 23 (1982) 181–192.
B.C. Eaves, “The linear complementarity problem”, Management Science 17 (1971) 612–634.
L.M. Kelly and L.T. Watson, “Erratum: some pertubation theorems for Q-matrices”, SIAM Journal on Applied Mathematics 34 (1978) 320–321.
C.E. Lemke, “Bimatrix equilibrium points and mathematical programming”, Management Science 11 (1965) 681–689.
C.E. Lemke, “On complementary pivot theory”, in: G.B. Dantzig and A.F. Veinott Jr., eds., Mathematics of the decision sciences, Part 1 (American Mathematical Society, Providence, RI, 1968) pp. 95–114.
O.L. Mangasarian, Nonlinear programming (McGraw-Hill, New York, 1969).
O.L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods”, Journal of Optimization Theory and Applications 22 (1977) 465–485.
O.L. Mangasarian, “Characterizations of linear complementarity problems as linear programs”, Mathematical Programming Study 7 (1978) 74–87.
O.L. Mangasarian, “Simplified characterizations of linear complementarity problems solvable as linear programs”, Mathematics of Operations Research 4 (1979) 268–273.
O.L. Mangasarian, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems”, Mathematical Programming 19 (1980) 200–212.
N. Megiddo and M. Kojima, “On the existence and uniqueness of solutions in nonlinear complementarity theory”, Mathematical Programming 12 (1977) 110–130.
K.G. Murty, “On the number of solutions to the complementarity problem and spanning properties of complementary cones”, Linear Algebra and its Applications 5 (1972) 65–108.
J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).
J.-S. Pang, “A note on an open problem in linear complementarity”, Mathematical Programming 13 (1977) 360–363.
J.-S. Pang, “On Q-matrices”, Mathematical Programming 17 (1979) 243–247.
S.M. Robinson, “Stability theory for systems of inequalities, Part I: Linear systems, Part II: Differentiable nonlinear systems”, SIAM Journal on Numerical Analysis 12 (1975) 754–769 and 13 (1976) 497–513.
S.M. Robinson, “Generalized equations and their solutions, Part I: Basic theory”, Mathematical Programming Study 10 (1979) 128–141.
S.M. Robinson, “Strongly regular generalized equations”, Mathematics of Operations Research 5 (1980) 43–62.
R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ 1970).
A.C. Williams, “Marginal values in linear programming”, Journal of the Society for Industrial and Applied Mathematics 11 (1963) 82–94.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
Mangasarian, O.L. (1982). Characterizations of bounded solutions of linear complementarity problems. In: Guignard, M. (eds) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120987
Download citation
DOI: https://doi.org/10.1007/BFb0120987
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00849-8
Online ISBN: 978-3-642-00850-4
eBook Packages: Springer Book Archive