# Continuity of the Solution Set of Homogeneous Equilibrium Problems and Linear Complementarity Problems

• Oettli Werner
• Yen Nguyen Dong
Chapter

## Abstract

Denote by S(M, q) the solution set of the linear complementarity problem
$$z \geqslant 0,{\mkern 1mu} Mz + q \geqslant 0,{\mkern 1mu} \left\langle {z,Mz + q} \right\rangle {\mkern 1mu} = 0,$$
where M ∈ ℝn×n and q ∈ ℝ n . 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 ∈ ℝ n . We prove that this property is characteristic for R 0-matrices. Part of our results extends to homogeneous equilibrium problems of the type
$$z{\mkern 1mu} \in K,{\mkern 1mu} f(z,y){\mkern 1mu} + {\mkern 1mu} \left\langle {q,y - z} \right\rangle \geqslant 0,{\mkern 1mu} \forall y \in {\mkern 1mu} K.$$

Here K ⊂ ℝ n is a closed convex cone and f: K × K → ℝ is such that fx, λy) = λρ+1 f(x,y) ∀x,yK, ∀λ ≥ 0, where ρ > 0 is a fixed constant.

## Key Words

Linear complementarity problem solution map upper semicontinuity nonlinear equilibrium problem

## References

1. [1]
E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems”, Math. Student 63, 123–145, 1994.
2. [2]
H. Brézis, “Analyse fonctionnelle”, Masson, Paris, 1983.Google Scholar
3. [3]
R.W. Cottle, J.-S. Pang, and R.E. Stone, “The Linear Complementarity Problem”, Academic Press, New York, 1992.
4. [4]
R.D. Doverspike, “Some perturbation results for the linear complementarity problem”, Math. Programming 23, 181–192, 1982.
5. [5]
C.B. Garcia, “Some classes of matrices in linear complementarity theory”, Math. Programming 5, 299–310, 1973.
6. [6]
M.S. Gowda, “On the continuity of the solution map in linear complementarity problems”, SIAM J. Optimization 2, 619–634, 1992.
7. [7]
M.S. Gowda, “Applications of degree theory to linear complementarity problems”, Math. Oper. Res. 18, 868–879, 1993.
8. [8]
M.S. Gowda and J.-S. Pang, “On solution stability of the linear complementarity problem”, Math. Oper. Res. 17, 77–83, 1992.
9. [9]
C.D. Ha, “Stability of the linear complementarity problem at a solution point”, Math. Programming 31, 327–338, 1985.
10. [10]
M.J.M. Jansen and S.H. Tijs, “Robustness and nondegenerateness for linear complementarity problems”, Math. Programming 37, 293–308, 1987.
11. [11]
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.
12. [12]
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.
13. [13]
K.G. Murty, “Linear Complementarity, Linear and Nonlinear Programming”, Heldermann-Verlag, Berlin, 1987.Google Scholar
14. [14]
S.M. Robinson, “Generalized equations and their solutions, Part I: Basic Theory”, Math. Programming Study 10, 128–141, 1979.
15. [15]
S.M. Robinson, “Some continuity properties of polyhedral multifunctions”, Math. Programming Study 14, 206–214, 1981.
16. [16]
R.T. Rockafellar, “Convex Analysis”, Princeton University Press, Princeton, 1970.

## Authors and Affiliations

• Oettli Werner
• 1
• 2
• Yen Nguyen Dong
• 2
1. 1.Fakultät für Mathematik und InformatikUniversität MannheimMannheimGermany
2. 2.Hanoi Institute of MathematicsBo Ho, HanoiVietnam