From optimization and variational inequalities to equilibrium problems. (English) Zbl 0888.49007

Blum, Eugen; Oettli, Werner

Math. Stud. 63, No. 1-4, 123-145 (1994).

The authors formulate the equilibrium problem consisting in finding\(\quad \bar x\in K\;\) such that \(f(\bar x,y)\geq 0\;\forall y\in K,\) where \(K\) is a given set and the function \(f:K\times K \to \mathbb R\) fulfils \(f(x,x)=0\;\forall x\in K.\) The problem contains as special cases variational inequalities, Nash equilibria in noncooperative games, complementarity and fixed point problems, convex differentiable optimization. The basic existence result for the equilibrium problem in the case \(f(x,y)=g(x,y)+h(x,y)\) is verified. The function \(g\) is monotone and satisfies a mild upper semicontinuity in \(x\) whereas \(h\) is not neccessarily monotone but has to satisfy a stronger u.s.c. condition in \(x.\) The existence theorem is modified for the case of monotone and maximal monotone operator \(g.\) The paper contains also the equilibrium problems over locally compact cones and complete metric spaces. It ends with variational principles for equilibrium problems.

Reviewer: I.Bock (Bratislava)

MSC:

49J40	Variational inequalities
49J27	Existence theories for problems in abstract spaces
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Keywords:

equilibrium problem; variational inequality; optimization; convex sets; fixed point; duality