$\varepsilon$-Optimality conditions for composed convex optimization problems

Abstract

The aim of the present paper is to provide a formula for the $\varepsilon$-subdifferential of $f+g\circ h$ different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Even more, using the $\varepsilon$-subdifferential formula we are able to derive necessary and sufficient conditions for the $\varepsilon$-optimal solutions of composed convex optimization problems.