Abstract
Denote by S(M, q) the solution set of the linear complementarity problem
where M ∈ ℝn×n and q ∈ ℝn. M is called an R 0-matrix iff S(M, 0) = {0}. Jansen and Tijs have proved that if M is an R 0-matrix, then the map S is upper semicontinuous at (M, q) for every q ∈ ℝn. We prove that this property is characteristic for R 0-matrices. Part of our results extends to homogeneous equilibrium problems of the type
Here K ⊂ ℝn is a closed convex cone and f: K × K → ℝ is such that f(λx, λy) = λρ+1 f(x,y) ∀x,y ∈ K, ∀λ ≥ 0, where ρ > 0 is a fixed constant.
Key Words
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems”, Math. Student 63, 123–145, 1994.
H. Brézis, “Analyse fonctionnelle”, Masson, Paris, 1983.
R.W. Cottle, J.-S. Pang, and R.E. Stone, “The Linear Complementarity Problem”, Academic Press, New York, 1992.
R.D. Doverspike, “Some perturbation results for the linear complementarity problem”, Math. Programming 23, 181–192, 1982.
C.B. Garcia, “Some classes of matrices in linear complementarity theory”, Math. Programming 5, 299–310, 1973.
M.S. Gowda, “On the continuity of the solution map in linear complementarity problems”, SIAM J. Optimization 2, 619–634, 1992.
M.S. Gowda, “Applications of degree theory to linear complementarity problems”, Math. Oper. Res. 18, 868–879, 1993.
M.S. Gowda and J.-S. Pang, “On solution stability of the linear complementarity problem”, Math. Oper. Res. 17, 77–83, 1992.
C.D. Ha, “Stability of the linear complementarity problem at a solution point”, Math. Programming 31, 327–338, 1985.
M.J.M. Jansen and S.H. Tijs, “Robustness and nondegenerateness for linear complementarity problems”, Math. Programming 37, 293–308, 1987.
O.L. Mangasarian and T.-H. Shiau, “Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems”, SIAM J. Control Optim. 25, 583–595, 1987.
K.G. Murty, “On the number of solutions to the complementarity problem and spanning properties of complementarity cones”, Linear Algebra Appl. 5, 65–108, 1972.
K.G. Murty, “Linear Complementarity, Linear and Nonlinear Programming”, Heldermann-Verlag, Berlin, 1987.
S.M. Robinson, “Generalized equations and their solutions, Part I: Basic Theory”, Math. Programming Study 10, 128–141, 1979.
S.M. Robinson, “Some continuity properties of polyhedral multifunctions”, Math. Programming Study 14, 206–214, 1981.
R.T. Rockafellar, “Convex Analysis”, Princeton University Press, Princeton, 1970.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this chapter
Cite this chapter
Werner, O., Dong, Y.N. (1995). Continuity of the Solution Set of Homogeneous Equilibrium Problems and Linear Complementarity Problems. In: Giannessi, F., Maugeri, A. (eds) Variational Inequalities and Network Equilibrium Problems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1358-6_17
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1358-6_17
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1360-9
Online ISBN: 978-1-4899-1358-6
eBook Packages: Springer Book Archive