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Zbl 1183.49012
Yao, Yonghong; Chen, Rudong; Xu, Hong-Kun
Schemes for finding minimum-norm solutions of variational inequalities. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 7-8, 3447-3456 (2010). ISSN 0362-546X

Summary: Consider the Variational Inequality (VI) of finding a point $x^*$ such that $$x^*\in\text{Fix}(T)\text{ and }\langle(I-S)x^*,x-x^*\rangle\ge 0,\quad x\in \text{Fix}(T)\tag*$$ 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 2000:: *49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators
47H09 Mappings defined by "shrinking" properties
65J15 Equations with nonlinear operators (numerical methods)

Keywords: variational inequality; nonexpansive mapping; iterative algorithm; implicit scheme; explicit scheme; fixed point; minimum norm

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