Farkas-type Results for Max-functions and Applications

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.

Farkas-type results for inequality systems with composed convex functions via conjugate duality

“…The approach we use is based on conjugate duality for an optimization problem consisting in minimizing the composition of an R k + -increasing and convex function with a convex vector function, subject to finitely many convex inequality constraints. The result we present generalizes some Farkas-type results presented by Boţ and Wanka in [4,5]. The connections between the Farkas-type results some theorems and of the alternative and, respectively, the theory of duality are exposed once more.…”

“…This special case of our initial problem is similar to the one studied in [5]. It is well known (see [8]) that the conjugate of the function…”

“…The definition of the operator T and relation (5) imply that there exist β 0 and two real numbers r 1 and r 2 such that r = r 1 + r 2 , while the pairs (β, r 1 ) and (p, r 2 ) are in epi(f * ) and epi((β T F ) * ), respectively. Thus…”

“…The construction of the dual is described here in detail and a constraint qualification assuring strong duality between the primal problem and its dual is introduced. Regarding the Fenchel-Lagrange dual problem, let us mention that this type of dual problem has been introduced, named and studied by Boţ and Wanka (see, for example, [2][3][4][5]17]). …”

“…Recently Boţ and Wanka have presented in [4,5] some Farkas-type results for inequality systems involving finitely many convex functions using an approach based on the theory of conjugate duality for convex problems. The aim of the present paper is to extend these results to convex optimization problems involving composed convex functions.…”