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 ). 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 the authors establish the equivalence between the variational inequality: , (the nonconvex variational inequality, NVI) and the fixed point problem using the projection operator technique. Here is a solution of the nonconvex variational inequality if and only if satisfies the relation: , where is the projection of (a real Hilbert space) onto the uniformly prox-regular set , 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 is pseudomonotone, and is a solution of NVI and is the approximate solution obtained from Algorithm 2 (For a given find the approximate solution by using the iterative schemes: one then has: , , and , if 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.