Dual Inequalities for Stabilized Column Generation Revisited

Timo Gschwind, Stefan Irnich
2016 INFORMS journal on computing  
Column generation (CG) models have several advantages over compact formulations, namely, they provide better LP bounds, may eliminate symmetry, and can hide non-linearities in their subproblems. However, users also encounter drawbacks in the form of slow convergency a.k.a. the tailing-off effect and the oscillation of the dual variables. Among different alternatives for stabilizing the CG process, Ben Amor et al. [Ben Amor, H., Desrosiers, J., and Valério de Carvalho, J. M. (2006) .
more » ... inequalities for stabilized column generation. Operations Research, 54(3), 454-463] suggest the use of dual-optimal inequalities (DOIs) in the context of cutting stock and bin packing problems. We generalize their results, provide new classes of (deep) DOIs, and show the applicability to other problems (vector packing, vertex coloring, bin packing with conflicts). We also suggest the dynamic addition of violated dual inequalities in a cutting-plane fashion and the use of dual inequalities that are not necessarily (deep) DOIs. In the latter case, a recovery procedure is needed to restore primal feasibility. Computational results proving the usefulness of the methods are presented. Acronym Full Name CG column generation RMP restricted master program DI dual inequality; any inequality in the variables of D DOI dual-optimal inequality; fulfilled by every dual-optimal solution of D DDOI (set of) deep dual-optimal inequality/ies WSI weighted subset inequality SI subset inequality CS cutting stock problem BP bin packing problem BPC bin packing problem with conflicts VP vector packing problem VC vertex coloring problem KP binary knapsack problem UKP unbounded knapsack problem KPC binary knapsack problem with conflicts DKP unbounded version of the d-dimensional knapsack problem MWIS maximum weight independent set Table 1: Some acronyms used in the paper set of DOIs, every single DI E j * π ≤ e j , where E j * refers to the jth row of E, must qualify as a DOI. This is fulfilled if all dual-optimal solutions π * ∈ D * respect E j * π * ≤ e j . The contribution of our paper consists in at least five new findings: First, we generalize Propositions 1 and 2 of Ben Amor et al. (2006) and Proposition 1 of Valério de Carvalho (2005) providing insights into the relations of optimal solution values and optimal solutions to models P , D,P , andD when E π ≤ e are DOIs or DDOIs. Second, we define several new classes of properties for integer valued matrices A and unit costs c = 1 that enable the identification of new DOIs and DDOIs. Herewith, we derive additional classes of DDOIs for BP and DOIs for CS. The latter can be extended also to the vector packing problem (VP) which can be seen as the multi-dimensional generalization of CS. The third aspect is the extension of the DI-based stabilization to several new classes of problems: We address the vertex coloring problem (VC) in which the task is to color the vertices of a given undirected graph with a minimum number of colors so that adjacent vertices receive different colors. The synthesis of VC and BP is the bin packing problem with conflicts (BPC). In all cases the derivation of DOIs and DDOIs is based on showing that the above mentioned new matrix properties are valid. Fourth, we show that a kind of over-stabilization of CG procedures can result in an overall faster convergency. There exist cases in which one can suspect that E π ≤ e are DDOIs, but this may actually not be true. The property may be fulfilled just for some instances of the problem or just for a proper subset that one is unable to identify. In the negative case, the outcome of the stabilized CG algorithm is an infeasible primal solution and a weaker bound than z P (see Proposition 1). We show that there is a constructive way of identifying the DIs that have cut off D * . This can be seen as a recovery procedure. In order to accelerate the overall CG approach, we therefore propose to alternate between solving the (over-)stabilized formulatioñ P and the recovery procedure until a feasible primal solution results. Fifth and finally, we show that the dynamic generation of DIs during the CG process is another option that often helps reducing the computation time. Note that up to now, DOIs and DDOIs have been used in a static fashion. Indeed, when the number of known DOIs or DDOIs is relatively small, e.g., polynomial in the size of the input instance, thenP is not significantly larger than P , and the advantage of stability can then outreach the resulting larger RMP re-optimization times. Such a family of DOIs has been used in a branch-and-price approach for the capacitated arc-routing problem (Bode and Irnich, 2012). However, in all problems considered here, the families of DIs, DOIs, and DDOIs are exponential in size. For the problems CS, BP, VC, and BPC we demonstrate that often the dynamic generation, sometimes in combination with an a priori addition of the expectedly strongest DIs provides the best trade-off between stabilization, effort of DI generation, and effort for re-optimizing the RMP. 3
doi:10.1287/ijoc.2015.0670 fatcat:nm74jopbhbf6bcgrxcf75o5x2q