Abstract
It has been shown by Lemke that if a matrix is copositive plus on ℝn, then feasibility of the corresponding linear complementarity problem implies solvability. In this article we show, under suitable conditions, that feasibility of ageneralized linear complementarity problem (i.e., defined over a more general closed convex cone in a real Hilbert space) implies solvability whenever the operator is copositive plus on that cone. We show that among all closed convex cones in a finite dimensional real Hilbert Space, polyhedral cones are theonly ones with the property that every copositive plus, feasible GLCP is solvable. We also prove a perturbation result for generalized linear complementarity problems.
Similar content being viewed by others
References
G. Allen, “Variational inequalities, complementarity problems, and duality theorems,”Journal of Mathematical Analysis and Applications 58 (1977) 1–10.
M.S. Bazaraa, J.J. Goode and M.Z. Nashed, “A nonlinear complementarity problem in mathematical programming in Banach space,”Proceedings of the American Mathematical Society 35 (1973) 165–170.
J.M. Borwein, “Alternative theorems for general complementarity problems,” in:Proceedings of the Cambridge Symposium on Infinite Programming. Lecture Notes in Economics and Mathematical Systems, Vol. 259 (Springer, Berlin, 1984) pp. 194–203.
J.M. Borwein, “Generalized linear complementarity problems treated without the fixed point theory,”Journal of Optimization Theory and Applications 43(3) (1984) 343–356.
A.T. Dash and S. Nanda, “A complementarity problem in mathematical programming in Banach space,”Journal of Mathematical Analysis and Applications 98 (1984) 318–331.
R. Doverspike, “Some perturbation results for the linear complementarity problem,”Mathematical Programming 23 (1982) 181–192.
I. Ekeland and R. Temam,Convex Analysis and Variational Problems (North-Holland American Elsevier, New York, 1976).
K. Fan, “A minimax inequality and applications,” in: O. Shisha, ed,Inequalities, Vol. III (Academic Press, New York, 1972) pp. 103–113.
M.S. Gowda, “Minimizing quadratic functionals over closed convex cones,”Bulletin of the Australian Mathematical Society 39(1) (1989) 15–20.
G.J. Habetler and A.L. Price, “Existence theory for generalized nonlinear complementarity problems,”Journal of Optimization Theory and Applications 7(4) (1971) 223–239.
G. Isac, “Nonlinear complementarity problem and Galerkin method,”Journal of Mathematical Analysis and Applications 108 (1985) 563–574.
S. Karamardian, “Generalized complementarity problem,”Journal of Optimization Theory and Applications 8(3) (1971) 161–168.
S. Karamardian, “The complementarity problem,”Mathematical Programming 2 (1972) 107–129.
S. Karamardian, “Complementarity problems over cones with monotone and pseudomonotone mappings,”Journal of Optimization Theory and Applications 18(4) (1976) 445–454.
S. Karamardian, “An existence theorem for the complementarity problem,”Journal of Optimization Theory and Applications 19(2) (1976) 227–232.
C.E. Lemke, “On complementary pivot theory,” in: G.B. Dantzig and A.F. Veinott Jr., eds.,Mathematics of the Decision Sciences, Part I (American Mathematical Society, Providence, RI, 1968).
G. Luna, “A remark on the nonlinear complementarity problem,”Proceedings of the American Mathematical Society 48 (1975) 132–134.
O.L. Mangasarian,Nonlinear Programming (McGraw-Hill, New York, 1969).
O.L. Mangasarian, “Characterizations of bounded solutions of linear complementarity problems,”Mathematical Programming Study 19 (1982) 153–166.
L. McLinden, “Stable monotone variational inequalities,” Technical Report #2734, Mathematics Research Center, University of Wisconsin (Madison, WI, 1984).
H. Mirkil, “New characterizations of polyhedral cones,”Canadian Journal of Mathematics IX(1) (1957) 1–4.
J.J. Moré, “Classes of functions and feasibility conditions in nonlinear complementarity problems,”Mathematical Programming 6 (1974) 327–338.
J.J. Moré, “Coercivity conditions in nonlinear complementarity problems,”SIAM Review 16(1) (1974) 1–16.
S. Nanda and S. Nanda, “A nonlinear complementarity problem in Banach space,”Bulletin of the Australian Mathematical Society 21 (1980) 351–356.
J. Parida and K.L. Roy, “An existence theorem for the nonlinear complementarity problem,”Indian Journal of Pure and Applied Mathematics 13(6) (1982) 615–619.
R. Saigal, “Extension of the generalized complementarity problem,”Mathematics of Operations Research 1(3) (1976) 260–266.
Author information
Authors and Affiliations
Additional information
This research has been partially supported by the Air Force Office of Scientific Research under grants #AFOSR-82-0271 and #AFOSR-87-0350.
Rights and permissions
About this article
Cite this article
Gowda, M.S., Seidman, T.I. Generalized linear complementarity problems. Mathematical Programming 46, 329–340 (1990). https://doi.org/10.1007/BF01585749
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585749