# 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

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© Springer Science+Business Media New York 1995

## 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