## An Introduction to Variational-like Inequalities

Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.4 Existence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.5 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

6.5.1 Auxiliary Principle Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.5.2 Proximal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

The theory of variational inequalities started with the pioneer work of Fichera [26] and Stampacchia [64], independently, related to Signori’s contact problem. Variational inequalities arise in models for a wide class of problems from science, social sciences, engineering, management, etc.; see, for example [2, 7, 8, 10, 13, 23, 30, 29, 32, 35, 36, 37, 41, 42, 43, 46, 47, 52, 58, 74] and the references therein. Because of its applications in branches of sciences, engineering, optimization, economics, equilibrium theory, etc. it has been extended and generalized in many different directions. It has been used as a tool to study different aspects of optimization problems; see, for example, [2, 23, 30, 28] and the references therein. Motivated by the application of variational inequalities to optimization problems, and the concept of invexity, Parida et al. [57] and Yang and Chen [72] independently replaced the linear term y − x appearing in the formulation of variational inequalities by a vector-valued term η(y, x), where η

is a vector-valued bifunction. Such variational inequality is called variationallike inequality or pre-variational inequality. Parida et al. [57] established some existence results for a solution of variational-like inequalities in the setting of the finite dimensional Euclidean space Rn by using the Kakutani fixedpoint theorem. They studied the relation between a variational-like inequality with a mathematical programming problem. Yang and Chen [72] introduced a new class of non-convex and non-smooth functions, called semi-preinvex functions. They derived the Fritz-John condition by using an alternative theorem for semi-preinvex program and studied the variational-like inequality. They also derived a necessary condition for an optimal solution of an optimization problem. Some existence theorems for solutions of a variational-like inequality were also proved. In 1994, Siddiqi et al. [62] and Ansari and Yao [3] studied variational-like inequalities in the setting of reflexive Banach spaces and topological vector spaces with or without convexity assumptions. Noor [53, 55] studied the relationship between a variational-like inequality problem and an optimization problem. He proved that the minimum of the arcwise directional differentiable semi-invex functions can be characterized by the class of variational-like inequalities. He also established an existence result for a solution of a variational-like inequality problem in the setting of Hilbert spaces and under the strong monotonicity and Lipschitz continuity assumptions of the underlying mappings.