General Variational Inequalities and Optimization

It is well known that general variational inequalities provide us with a unified, natural, novel, and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, some new classes of general variational inequalities are introduced and analyzed, which are related to the optimality criteria of the differentiable of nonconvex functions. A number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques include projection, Wiener-Hopf equations, dynamical systems, auxiliary principle, and penalty function. Convergence analysis of these methods is investigated under suitable conditions. Some numerical examples are given to illustrate and implement the proposed methods: merit functions, sensitivity analysis, well-posed criteria, error bounds, and other aspects of these general variational inequalities. Several new classes of the nonconvex functions such as exponentially convex functions, log $$\log $$ -convex function, and kg-convex functions are introduced, which have applications in information, data analysis, and machine learning. It is shown that the optimality conditions of these new classes of nonconvex functions motivated us to study some new classes of general variational inequalities. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving these new general variational inequalities: higher-order general variational inequalities. Our proofs of convergence are very simple compared to other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Several open problems such as error bounds, sensitivity analysis, and well-posed criteria of the proposed new classes of general variational inequalities have been suggested for further research in these areas. It is expected that the ideas and techniques of this paper may inspire the readers to explore the applications of these results in solving complicated problems, which arise in various branches of pure and applied sciences.