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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.
Dinh, Nguyen ; Goberna, Miguel A. ; López, Marco A. ; Son, Ta Quang 1
@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},
pages = {580--597},
publisher = {EDP-Sciences},
volume = {13},
number = {3},
year = {2007},
doi = {10.1051/cocv:2007027},
mrnumber = {2329178},
zbl = {1126.90059},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007027/}
}
TY - JOUR AU - Dinh, Nguyen AU - Goberna, Miguel A. AU - López, Marco A. AU - Son, Ta Quang TI - New Farkas-type constraint qualifications in convex infinite programming JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 580 EP - 597 VL - 13 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007027/ DO - 10.1051/cocv:2007027 LA - en ID - COCV_2007__13_3_580_0 ER -
%0 Journal Article %A Dinh, Nguyen %A Goberna, Miguel A. %A López, Marco A. %A Son, Ta Quang %T New Farkas-type constraint qualifications in convex infinite programming %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 580-597 %V 13 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv:2007027/ %R 10.1051/cocv:2007027 %G en %F 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, Tome 13 (2007) no. 3, pp. 580-597. doi: 10.1051/cocv:2007027
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