This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.

Classification: 90C25, 90C34, 90C46, 90C48

Keywords: convex infinite programming, KKT and saddle point optimality conditions, duality theory, Farkas-type constraint qualification

@article{COCV_2007__13_3_580_0, author = {Dinh, Nguyen and Goberna, Miguel A. and L\'opez, Marco A. and Son, Ta Quang}, title = {New Farkas-type constraint qualifications in convex infinite programming}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {13}, number = {3}, year = {2007}, pages = {580-597}, doi = {10.1051/cocv:2007027}, zbl = {1126.90059}, mrnumber = {2329178}, language = {en}, url = {http://www.numdam.org/item/COCV_2007__13_3_580_0} }

Dinh, Nguyen; Goberna, Miguel A.; López, Marco A.; Son, Ta Quang. New Farkas-type constraint qualifications in convex infinite programming. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, pp. 580-597. doi : 10.1051/cocv:2007027. http://www.numdam.org/item/COCV_2007__13_3_580_0/

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