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Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming. (English) Zbl 1206.90198
The paper is centered around the Farkas lemma and its extensions. The authors study several constraint qualifications for optimization problems with constraints given by an inequality system defined by a possibly infinite family of proper functions. Several examples of problems that recast into the form of the optimization problems above mentioned are showed in the introduction. For example, linear semi-infinite optimization problems, best approximation with restricted ranges, conic programming problems, the convex composite problem. The paper is focused on two aspects: the extended Farkas lemma and the strong Lagrangian duality. The authors use constraint qualifications involving epigraphs to provide complete characterizations for the Farkas rule and the stable Farkas rule and for the (strong/strong stable) Lagrangian duality. Several known results for the conic programming problem are extended.

##### MSC:
 90C34 Semi-infinite programming 90C25 Convex programming 52A07 Convex sets in topological vector spaces (aspects of convex geometry) 41A29 Approximation with constraints 90C46 Optimality conditions and duality in mathematical programming
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