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

Abstract

We present some Farkas-type results for inequality systems involving finitely many functions. Therefore we use a conjugate duality approach applied to an optimization problem with a composed convex objective function and convex inequality constraints. Some recently obtained results are rediscovered as special cases of our main result.

Some Farkas-type results for fractional programming problems with DC functions

“…. , f m : R n → R, the infimal convolution function ( f 1 · · · f m ) : R n → R is defined by 4,14]). Let f 1 , .…”

“…, f m : R n → R be proper convex functions. If the set m i=1 ri(dom( f i )) is nonempty, then 4]). Let f : R n → R be a proper function and α > 0 a real number.…”

ɛ-Optimality conditions for composed convex optimization problems

“…(c) In case ε = 0 by means of (i)-(iii) we rediscover the optimality conditions given in the past for characterizing the (exact) optimal solutions of the primal problem (P) and its dual problem (D) (see, for example, [3])…”

“…Therefore, in order to be able to characterize the approximate solutions of an optimization problem involving composed convex functions, it is important to provide a formula for the ε-subdifferential of a composed convex function (the interested reader can consult the papers [1,3,4,[9][10][11]13] for more information regarding the optimization problems involving composed convex functions).…”

A Fenchel–Lagrange Duality Approach for a Bilevel Programming Problem with Extremal-Value Function

“…In this paper, adopting the Fenchel-Lagrange duality, which has been first introduced by Boţ and Wanka in [14] for ordinary convex programming problems, we give the so-called Fenchel-Lagrange dual of the bilevel problem (S). Based on some results given in [15][16][17][18], we first show that a strong duality holds between them. Then, we provide necessary and sufficient optimality conditions for (S) and its dual.…”