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Schemes for finding minimum-norm solutions of variational inequalities. (English) Zbl 1183.49012
Summary: Consider the Variational Inequality (VI) of finding a point $$x^*$$ such that
$x^*\in\text{Fix}(T)\text{ and }\langle(I-S)x^*,x-x^*\rangle\geq 0,\quad x\in \text{Fix}(T)\tag{*}$
where $$T,S$$ are nonexpansive self-mappings of a closed convex subset $$C$$ of a Hilbert space, and $$\text{Fix}(T)$$ is the set of fixed points of $$T$$. Assume that the solution set $$\Omega$$ of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI $$(*)$$; namely, the unique solution $$x^*$$ to the quadratic minimization problem: $$x^*=\text{argmin}_{x\in \Omega}\|x\|^2$$.

##### MSC:
 49J40 Variational inequalities 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 65J15 Numerical solutions to equations with nonlinear operators
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