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

T. Q. Son, J. J. Strodiot () and V. H. Nguyen
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T. Q. Son: Nhatrang Teacher College
J. J. Strodiot: University of Namur (FUNDP)
V. H. Nguyen: University of Namur (FUNDP)

Journal of Optimization Theory and Applications, 2009, vol. 141, issue 2, No 10, 389-409

Abstract: 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.

Keywords: Karush-Kuhn-Tucker conditions up to ε; Approximate solutions; Quasisaddlepoints; ε-Lagrange duality (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10957-008-9475-2

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