Differentiable non-convex functions and general variational inequalities

In this paper, we introduce a new class of non-convex functions, which is called the g-convex functions. We prove that the minimum of the differentiable non-convex functions on the non-convex sets can be characterized by a class of variational inequalities, which is called the general variational inequality. Essentially using the projection technique, we establish the equivalence between the general variational inequalities and the fixed-point problems as well as with the Wiener–Hopf equations. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general variational inequalities. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.