This paper provides new Farkas-type results characterizing the inclusion A ⊂ B, where A and B are subsets of a locally convex space X. The sets A and B are described here by means of vector functions from X to other locally convex spaces Y (equipped with the partial ordering associated with a given convex cone K ⊂ Y) and Z, respectively. These new Farkas lemmas are obtained via the complete characterization of the K-epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the autors [J. Optim. Theory Appl. 173, 357-390 (2017)], the characterizations of A ⊂ B are expressed here in terms of the data.

We characterize the subdifferential of the supremum function of finitely and infinitely indexed families of convex functions. The main contribution of this paper consists of providing formulas for such a subdifferential under weak continuity assumptions. The resulting formulas are given in terms of the exact subdifferential of the data functions at the reference point, and not at nearby points as in [Valadier, C. R. Math. Acad. Sci. Paris, 268 (1969), pp. 39-42]. We also derive new Fritz John-and KKT-type optimality conditions for semi-infinite convex optimization, omitting the continuity/closedness assumptions in [Dinh et al., ESAIM Control Optim. Calc. Var., 13 (2007), pp. 580-597]. When the family of functions is finite, we use continuity conditions concerning only the active functions, and not all the data functions as in [Rockafellar,

The objective of this paper is to analyse under what well-known operations the class of quasipolyhedral convex functions, which can be regarded as an extension of that of polyhedral convex functions, is closed. The operations that will be considered are those that preserve polyhedral convexity, such that the image and the inverse image under linear transformations, right scalar multiplication (including the case where = 0 + ) and pointwise addition.

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