Journal of Convex Analysis 13 (2006), No. 1, 113--133
Copyright Heldermann Verlag 2006
From Linear to Convex Systems: Consistency, Farkas' Lemma and Applications
N. Dinh
Dept. of Mathematics-Informatics, University of Pedagogy, Ho Chi Minh City, Vietnam
Miguel A. Goberna
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain
Marco Antonio López
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain
This paper analyzes inequality systems with an arbitrary number of proper lower semicontinuous convex constraint functions and a closed convex constraint subset of a locally convex topological vector space. More in detail, starting from well-known results on linear systems (with no constraint set), the paper reviews and completes previous works on the above class of convex systems, providing consistency theorems, two new versions of Farkas' lemma, and optimality conditions in convex optimization. A new closed cone constraint qualification is proposed. Suitable counterparts of these results for cone-convex systems are also given.
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