  Augmented Lagrangian dual augments the classical Lagrangian dual with a
non-negative non-linear penalty function of the violation of the
relaxed/dualized constraints in order to reduce the duality gap. We investigate
the cases in which mixed integer convex optimization problems have an exact
penalty representation using sharp augmenting functions (norms as augmenting
penalty functions). We present a generalizable constructive proof technique for
proving existence of exact penalty representations for mixed integer convex
programs under specific conditions using the associated value functions. This
generalizes the recent results for MILP (Feizollahi, Ahmed and Sun, 2017) and
MIQP (Gu, Ahmed and Dey 2020) whilst also providing an alternative proof for
the aforementioned along with quantification of the finite penalty parameter in
these cases.
