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Zbl 1211.65082
Aslam Noor, Muhammad; Al-Said, Eisa; Noor, Khalida Inayat; Yao, Yonghong
Extragradient methods for solving nonconvex variational inequalities. (English)
[J] J. Comput. Appl. Math. 235, No. 9, 3104-3108 (2011). ISSN 0377-0427

This paper is devoted to the study of a new class of variational inequalities (the nonconvex variational inequalities), and for a new class of nonconvex sets (uniformly prox-regular set $K_r$). This class of uniformly prox-regular sets have played an important part in many nonconvex applications (optimization, dynamic systems and differential inclusions). For a nonlinear operator $T$ the authors establish the equivalence between the variational inequality: $<Tu, v-u> \geq 0$, $[u,v] \in K_r$ (the nonconvex variational inequality, NVI) and the fixed point problem using the projection operator technique. Here $u \in K_r$ is a solution of the nonconvex variational inequality if and only if $u \in K_r$ satisfies the relation: $u=P_{K_r} [u - \rho T u]$, where $P_{K_r}$ is the projection of $H$ (a real Hilbert space) onto the uniformly prox-regular set $K_r$, which implies that NVI is equivalent to the fixed point problem. This equivalent formulation is used to suggest and analyze the implicit iterative method for solving NVI. If the operator $T$ is pseudomonotone, and $ u \in K_r$ is a solution of NVI and $u_{n+1}$ is the approximate solution obtained from Algorithm 2 (For a given $u_\diamond \in H$ find the approximate solution $u_{n+1}$ by using the iterative schemes: $u_{n+1} = P_{K_r} [u_n - \rho T u_{n+1}], n = 0,1, ...)$ one then has: $\|u-u_{n+1}\|^2 \leq \|u - u_n\|^2 - \|u_{n+1} - u_n \|^2$, $\rho >0$, and $\lim_{n \rightarrow \infty} u_n = u$, if $H$ is a finite dimensional space. \par The authors use the idea of Noor to prove that the convergence of the extragradient method requires only pseudo-monotonicity, which is a weaker condition than monotonicity. Thus proposed result represents an improvement and refinement of the known results.
[Jan Lov\'\išek (Bratislava)]

MSC 2000:: *65K15
49J40 Variational methods including variational inequalities
90C33 Complementarity problems

Keywords: Variational inequalities; Monotone operators; Iterative method; Projection operator; Convergence; Algorithm

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