Journal
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
Volume 13, Issue 3, Pages 580-597Publisher
EDP SCIENCES S A
DOI: 10.1051/cocv:2007027
Keywords
convex infinite programming; KKT and saddle point optimality conditions; duality theory; Farkas-type; constraint qualification
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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.
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