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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:
90C34Semi-infinite programming
90C25Convex programming
52A07Convex sets in topological vector spaces (convex geometry)
41A29Approximation with constraints
90C46Optimality conditions, duality