Sequential Optimality Conditions and Variational Inequalities

In recent years, sequential optimality conditions are frequently used for convergence of iterative methods to solve nonlinear constrained optimization problems. The sequential optimality conditions do not require any of the constraint qualications. In this paper, We present the necessary sequential complementary approximate Karush Kuhn Tucker (CAKKT) condition for a point to be a solution of a nonlinear optimization problem. The nonlinear optimization problem is associated with the variational inequality problem. We also extend the complementary approximate Karush Kuhn Tucker condition from scalar optimization problem to multiobjective optimization problem and associated with the vector variational inequality problem. Further, we prove that with some extra conditions of convexity and affinity, complementary approximate Karush Kuhn Tucker conditions are sufficient for the variational inequality problem and vector variational inequality problem. Finally, we verify our results via illustrative examples. An example shows that a point which is a solution of variational inequality problem is also a CAKKT point.