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Extragradient methods for solving nonconvex variational inequalities. (English) Zbl 1211.65082

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: $\ge 0$, $\left[u,v\right]\in {K}_{r}$ (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: $u={P}_{{K}_{r}}\left[u-\rho Tu\right]$, 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: ${u}_{n+1}={P}_{{K}_{r}}\left[{u}_{n}-\rho T{u}_{n+1}\right],n=0,1,···\right)$ one then has: $\parallel u-{u}_{n+1}{\parallel }^{2}\le \parallel u-{u}_{n}{\parallel }^{2}-{\parallel {u}_{n+1}-{u}_{n}\parallel }^{2}$, $\rho >0$, and ${lim}_{n\to \infty }{u}_{n}=u$, if $H$ is a finite dimensional space.

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.

##### MSC:
 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)