On the augmented Lagrangian dual for integer programming

Abstract

We consider the augmented Lagrangian dual for integer programming, and provide a primal characterization of the resulting bound. As a corollary, we obtain proof that the augmented Lagrangian is a strong dual for integer programming. We are able to show that the penalty parameter applied to the augmented Lagrangian term may be placed at a fixed, large value and still obtain strong duality for pure integer programs.

Combining penalty‐based and Gauss–Seidel methods for solving stochastic mixed‐integer problems

“…In Boland and Eberhard (), and later in Feizollahi et al. (), it has been observed that a zero duality gap is achievable for dual problems based on an augmented Lagrangian in MIP problems.…”

“…Recent results have shown that, for penalty functions satisfying some slight conditions, the augmented Lagrangian dual is capable of asymptotically achieving zero duality gap when the penalty term coefficient ρ is allowed to go to infinity (Boland and Eberhard, , Proposition 3; Feizollahi et al., , Proposition 2). However, despite the theoretical relevance of this observation, it is not practically meaningful to deal with large‐valued penalty parameters, in large part due to the associated numerical issues that arise.…”

“…Furthermore, Boland and Eberhard (, Corollary 1) and (Feizollahi et al., , Theorem 4) demonstrate that it is possible to circumvent this drawback if the augmentation of the Lagrangian dual is made using a norm as the penalty function. In this case, the theory suggests that it is possible to attain strong duality for a finite value of ρ.…”

“…Remark In Boland and Eberhard (), other conditions that do not require the assumption of rationality of the data defining the problem are given that also ensure a limiting zero duality gap.…”

“…In this paper, we propose an alternative approach to deal with SMIP problems that build upon recent theoretical results from Boland and Eberhard () and Feizollahi et al. () showing that duality gaps can be diminished with the use of finite‐valued penalties for a specific class of penalty functions.…”