Abstract not found

We present some new Farkas-type results for inequality systems involving a finite as well as an infinite number of convex constraints. For this, we use two kinds of conjugate dual problems, namely an extended Fenchel-type dual problem and the recently introduced Fenchel-Lagrange dual problem. For the latter, which is a ”combination” of the classical Fenchel and Lagrange duals, the strong duality is established.

Abstract We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature. Keywords Conjugate functions · Fenchel-Lagrange duality · Composed convex optimization problems · Cone constraint qualifications

Abstract. We present some Farkas-type results for inequality systems involving finitely many convex constraints as well as convex max-functions. Therefore we use the dual of a minmax optimization problem. The main theorem and its consequences allows us to establish, as particular instances, some set containment characterizations and to rediscover two famous theorems of the alternative. Keywords: duality, Farkas-type results, minmax programming, set containment, theorems of the alternative AMS subject classification: 49N15, 90C25, 90C46 1.