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

In this paper epsilon-optimality conditions are given for a nonconvex programming problem which has an infinite number of convex constraints. In a first part we introduce the concept of regular "-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to epsilon and obtained by weakening the classical complementarity conditions. Then, thanks to the Ekeland Variational Principle, we first show that these KKT conditions are necessary for at least an almost regular epsilon-solution without any constraint qualification and then that they are also sufficient for epsilon-optimality when the objective function is epsilon-semiconvex. In a second part we define quasi saddle-points associated with an epsilon-Lagrangian functional and we investigate their relations with generalized KKT conditions. In particular, we formulate a Wolfe type dual problem which allows us to present epsilon-duality theorems and relations between KKT conditions and regular epsilon-solutions for the dual. Finally we apply these results to two important semi-infinite programming problems: the coneconstrained convex problem and the semi-definite programming problem.