Formulas for Asymptotic Functions via Conjugates, Directional Derivatives and Subdifferentials

The q-asymptotic function is a new tool that permits to study nonconvex optimization problems with unbounded data. It is particularly useful when dealing with quasiconvex functions. In this paper, we obtain formulas for the q-asymptotic function via c-conjugates, directional derivatives and subdifferentials. We obtain them under lower semicontinuity or local Lipschitz assumptions. The well-known formulas for the asymptotic function in the convex case are consequences of these ones. We obtain a new formula for the convex case.