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Zbl 1218.90155
Dinh, N.; Nghia, T.T.A.; Vallet, G.
A closedness condition and its applications to DC programs with convex constraints. (English)
[J] Optimization 59, No. 3-4, 541-560 (2010). ISSN 0233-1934; ISSN 1029-4945/e

Summary: This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called ``Toland-Fenchel-Lagrange'' duality theorems. As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived. Then, we establish optimality conditions for DC problem under convex constraints. Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well. Most of the results are established under the (CC) condition. This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs.

MSC 2000:: *90C25 Convex programming
90C26 Nonconvex programming
90C46 Optimality conditions, duality
49K30 Optimal solutions belonging to restricted classes

Keywords: DC programs; closedness conditions; closed-cone constraint qualification; Farkas lemmas; Fenchel-Lagrange duality; Toland-Fenchel-Lagrange duality

Cited in: Zbl 1228.90078

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