On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations

Strengthening linear relaxations and the bounds of mixed integer linear programs has been an active research topic for decades. Enumeration-based methods for integer programming like linear programming-based branch-and-bound exploit strong dual bounds to discard unpromising regions of the feasible space. In this paper, we consider the strengthening of linear programs via a composite of Dantzig-Wolfe and Fenchel decompositions. We provide geometric interpretations of these two standard methods. Motivated by these geometric interpretations, we introduce a novel approach for solving Fenchel sub-problems and introduce a novel algorithmic method originally combining Dantzig-Wolfe and Fenchel decompositions. We carry out extensive computational experiments assessing the performance of the novel method on the unsplittable flow problem. This new approach gives very promising results when compared to usual decomposition methods. • We introduce a novel approach to the Fenchel sub-problem when a directional normalization is used. The proposed method possesses reduces the numerical instabilities of a direct resolution approach commonly used. We show that the new approach solves the Fenchel sub-problem in finitely many iterations. • We introduce a new decomposition method inspired by both the Dantzig-Wolfe and the Fenchel decompositions. The method uses a Dantzig-Wolfe master problem and a Fenchel master problem. A Fenchel sub-problem guided with a directional normalization is used to coordinate the two master problems. • Although the theory behind the new method is general, we expect this method to outperform the Dantzig-Wolfe decomposition mainly when the latter suffers from convergence issues due to degeneracy. To highlight this we conduct an extensive computational study on the unsplittable flow problems for which the Dantzig-Wolfe decomposition is known to suffer from convergence issues. The new method is shown to perform well; especially on instances presenting high degrees of degeneracy. We provide a possible explanation of the resilience of our method to degeneracy based on our findings.