Abstract
This paper is intended to provide an updated survey of recent optimality theory for infinite-dimensional convex programming. It aims at establishing theoretical support for algorithmic developments. Two alternative strategies inspire the approaches presented in the paper. The first one consists of replacing the family of constraints by a single one, appealing to the supremum function, and is based on various characterizations of the subdifferential of the pointwise supremum of convex functions. The second one uses appropriate characterizations of affine consequent inequalities of the constraint system exploiting ad hoc constraint qualifications.
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