Some iterative methods for nonconvex variational inequalities

Abstract

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener‐Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener‐Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques. Variational inequalities theory, which was introduced by Stampacchia [35], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. During the last three decades, there has been considerable activity in the development of numerical techniques for solving variational inequalities. There is a substantial number of numerical methods, including projection method and its variant forms, Wiener‐Hopf equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems; see [1‐35] and the references therein. It is worth mentioning that almost all the results regarding the existence of and iterative schemes for variational inequalities that have been investigated and considered are for the case where the underlying set is a convex set. This is because all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general when the sets are nonconvex. Noor [25, 28] has introduced and studied a new class of variational inequalities, which is called the nonconvex variational inequality in conjunction with the uniformly prox-regular sets, which are nonconvex and include the convex sets as a special case, see [7, 33]. Noor [25, 28] has shown that the projection technique can be extended for the nonconvex variational inequalities and has established the equivalence between the nonconvex variational inequalities and fixed point problems using essentially the projection technique. This equivalent alternative formulation is used to discuss the existence of a solution of the nonconvex variational inequalities, which is Theorem 3.1. We use this alternative equivalent formulation to suggest and analyze an implicit type iterative method for solving the nonconvex variational inequalities. In order to implement this new implicit method, we use the predictor-corrector technique to suggest a two-step method for solving the nonconvex variational inequalities, which is Algorithm 3.4. We also consider the convergence (Theorem 3.2) of the new iterative method under some suitable conditions. We have also suggested three-step iterative methods for solving nonconvex variational inequalities. Some special cases are also discussed. We also introduce and consider the problem of solving the Wiener‐Hopf equations. Using essentially the projection technique, we show that the nonconvex variational inequalities are equivalent to the Wiener‐Hopf equations. This alternative equivalent formulation is more general and flexible than the projection operator technique. This alternative equivalent formulation is used to suggest and analyze a number of iterative methods for solving the nonconvex variational inequalities. These iterative methods are the subject of Section 4. We also consider the convergence criteria of the proposed iterative methods under some suitable conditions. Several special cases are