Exact Augmented Lagrangian Duality for Mixed Integer Convex Optimization

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1622-1645, June 2024.
Abstract. Augmented Lagrangian dual augments the classical Lagrangian dual with a nonnegative nonlinear 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 mixed integer linear programming [M. J. Feizollahi, S. Ahmed, and A. Sun, Math. Program., 161 (2017), pp. 365–387] and mixed integer quadratic progamming [X. Gu, S. Ahmed, and S. S. Dey, SIAM J. Optim., 30 (2020), pp. 781–797] while also providing an alternative proof for the aforementioned along with quantification of the finite penalty parameter in these cases.