Farkas Lemma for Convex Systems Revisited and Applications to Sublinear-Convex Optimization Problems

Abstract

In this paper we establish new versions of the Farkas lemma for systems which are convex with respect to a cone and convex with respect to an extended sublinear function under some Slater-type constraint qualification conditions and in the absence of lower semi-continuity and closedness assumptions on the functions and constrained sets. The results can be considered as counterparts of some of the earlier corresponding results in Dinh et al. (SIAM J. Optim. 24: 678–701, 2014). As consequences, we get extensions of the Hahn–Banach theorem for extended sublinear functions (the situation where the celebrated Hahn–Banach theorem failed). The results obtained are then applied to provide duality results and optimality conditions for a class of composite problems involving sublinear-convex mappings. Two special cases are examined at the end of the paper. In the first one our results give rise to some generalized Fenchel duality theorems while in the second one, in normed spaces, our result leads to the separation theorem for convex sets (not necessarily closed) in normed spaces.