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: \(x^*=\text{argmin}_{x\in \Omega}\|x\|^2\).


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
Full Text: DOI


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