ε-Optimality and ε-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints

In this paper, epsilon-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce the concept of regular epsilon-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to epsilon and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming any constraint qualification. Then, we prove that these conditions are also sufficient for epsilon-optimality when the constraints are convex and the objective function is epsilon-semiconvex. In a second part, we define quasisaddlepoints associated with an epsilon-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate a Wolfe-type dual problem which allows us to present epsilon-duality theorems and relationships between the KKT conditions and regular epsilon-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained convex problem and the semidefinite programming problem.