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Zbl 1215.49020
Yao, Yonghong; Liou, Yeong-Cheng; Kang, Shin Min
Algorithms construction for variational inequalities.
(English)
[J] Fixed Point Theory Appl. 2011, Article ID 794203, 12 p. (2011). ISSN 1687-1812/e

Summary: We devote this paper to solving the Variational Inequality (VI) of finding $x^*$ with property $x^*\in \text{Fix}(T)$ such that $\langle(A-\gamma f)x^*,x-x^*\rangle\ge 0$ for all $x\in \text{Fix}(T)$. Note that this hierarchical problem is associated with some convex programming problems. For solving the above VI, we suggest two algorithms: Implicit Algorithm: $x_t= TPc[I-t(A-\gamma f)]x_t$ for all $t\in(0,1)$ and Explicit Algorithm: $x_{n+1}= \beta_nx_n+(1-\beta_n) TPc[1-\alpha_n(A-\gamma f)]x_n$ for all $n\ge 0$ . It is shown that these two algorithms converge strongly to the unique solution of the above VI. As special cases, we prove that the proposed algorithms strongly converge to the minimum norm fixed point of $T$.
MSC 2000:
*49J40 Variational methods including variational inequalities
90C25 Convex programming

Keywords: variational inequality; hierarchical problem; convex programming; minimum norm fixed point

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