A Customized Augmented Lagrangian Method for Block-Structured Integer Programming

Rui Wang, Chuwen Zhang, Shanwen Pu, Jianjun Gao, Zaiwen Wen

Abstract:

Integer programming with block structures has received considerable attention recently and is widely used in many
practical applications such as train timetabling and vehicle routing
problems. It is known to be NP-hard due to the presence of integer
variables. We define a novel augmented Lagrangian function by
directly penalizing the inequality constraints and establish the strong duality between the primal problem and the augmented Lagrangian dual problem. Then, a customized augmented Lagrangian method is proposed to address the block-structures. In particular, the minimization of the augmented Lagrangian function is decomposed into multiple subproblems by decoupling the linking
constraints and these subproblems can be efficiently solved using the block coordinate descent method. We also establish the convergence property of the proposed method. To make the algorithm more practical, we further introduce several refinement techniques
to identify high-quality feasible solutions. Numerical experiments
on a few interesting scenarios show that our proposed algorithm
often achieves a satisfactory solution and is quite effective.