On some generalization of Karamardian’s theorem on the complementarity problem. (English) Zbl 0653.90076

Assume that \(<E,F>\) is a dual system of locally convex spaces, \(K\subset E\) is a closed convex cone, K * is the dual of K that is \(\{\) \(y\in F:\) \(<x,y>\geq O\), \(x\in K\}\) and f: \(K\to F\) is a mapping. The complementarity problem associated with K and f is: find \(x_*\in K\) such that \(f(x_*)\in K\) * and \(<x_*,f(x_*)>=O\). The paper presents a generalization of Karamardian’s theorem (concerning the existence of solutions) improving previous incorrect results. A new variant of the theorem in Hilbert space where K is a Galerkin cone, is given.
Reviewer: A.L.Dontchev


90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)