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Projection methods, isotone projection cones, and the complementarity problem. (English) Zbl 0711.47030
Let \((H,<.,.>)\) be a Hilbert space and \(K\subset H\) a closed convex cone. Let \(K^*\) be the dual cone of \(K\), that is \(K^*=\{y\in H|<x,y>\geq 0; \forall x\in K\}.\)
Given the mappings \(f,g: K\to H\), we consider the explicit and the implicit complementarity problems defined respectively by
\(\text{E.C.P.}(f,{\mathbb{K}})\): find \(x_0\in K\) such that \(f(x_0)\in K^*\) and \(<x_0,f(x_0)>=0\) and
\(\text{I.C.P.}(f,g,{\mathbb{K}})\): find \(x_ 0\in K\)such that \(g(x_0)\in K\), \(f(x_0)\in K^*\) and \(<g(x_0),f(x_0)>=0.\)
Supposing that \(K\) is an isotone projection cone, that is, it has the property: \(y-x\in K\) implies \(P_ K(y)-P_ K(x)\in K\) for every \(x,y\in H\) (where \(P_K\) is the projection operator) we describe and we study several iterative methods for solving the problems \(\text{E.C.P.}(f,K)\) and \(\text{I.C.P.}(f,g,K)\).
Reviewer: G.Isac

MSC:
47J05 Equations involving nonlinear operators (general)
49J27 Existence theories for problems in abstract spaces
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