Abstract
The well-known Farkas--Minkowski (FM) type qualification plays an important role in linear semi-infinite programming and has been extensively developed by many authors in establishing optimality conditions, duality, and stability for semi-infinite programming. In this paper, we introduce the concept of the quasi-Slater condition for a semi-infinite convex inequality system and present that the Slater type conditions imply the FM qualification under some appropriate continuity assumption of the set-valued mapping $i\mapsto f_i(x)$. Applying these relationships, we establish dual characterizations, both asymptotic and nonasymptotic, for set containment problems and provide some sufficient conditions for ensuring the strong Lagrangian duality and Farkas lemma.
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