Abstract
In this paper, ε-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 ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε 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 ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-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 ε-duality theorems and relationships between the KKT conditions and regular ε-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.
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Communicated by J.P. Crouzeix.
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Son, T.Q., Strodiot, J.J. & Nguyen, V.H. ε-Optimality and ε-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints. J Optim Theory Appl 141, 389–409 (2009). https://doi.org/10.1007/s10957-008-9475-2
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DOI: https://doi.org/10.1007/s10957-008-9475-2