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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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