Abstract We report on experiments with turning the branch-price-andcut framework SCIP into a generic branch-price-and-cut solver. That is, given a mixed integer program (MIP), our code performs a Dantzig-Wolfe decomposition according to the user’s specification, and solves the resulting re-formulation via branch-and-price. We take care of the column generation subproblems which are solved as MIPs themselves, branch and cut on the original variables (when this is appropriate), aggregate identical subproblems, etc. The charm of building on a well-maintained framework lies in avoiding to re-implement state-of-the-art MIP solving features like pseudo-cost branching, preprocessing, domain propagation, primal heuristics, cutting plane separation etc. 1

Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered block-diagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proof-of-concept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 0-1 dynamic knapsack problem (without block-diagonal structure) and on general MIPLIB2003 instances. Our results support that Dantzig-Wolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.

In many mixed integer programs there is some embedded problem structure which can be exploited, often by a decomposition. When the relaxation in each node of a branch-andbound tree is solved by column generation, one speaks of branch-and-price. Optionally, cutting planes can be added in order to strengthen the relaxation, and this is called branch-price-and-cut. We introduce the common concepts of convexification and discretization to arrive at a Dantzig-Wolfe type reformulation of a mixed integer program. The relation between the original and the extended formulations helps us understand how cutting planes should be formulated and how branching decisions can be taken while keeping the column generation subproblems manageable.

Abstract not found

Dantzig-Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the process can be automated. In particular the detection (better: the construction) of a matrix structure that is useful for Dantzig-Wolfe reformulation of a MIP can be accomplished by suitably permuting rows and columns. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the reformulated problem exploiting column generation. Our results support that Dantzig-Wolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.

We provide an overview of our recent efforts to automatize Dantzig-Wolfe reformulation and column generation/branch-and-price for structured, large-scale integer programs. We present the need for and the benefits from a generic implementation which does not need any user input or expert knowledge. A focus is on detecting structures in integer programs which are amenable to a Dantzig-Wolfe reformulation. We give computational results and discuss future research topics.

Dantzig-Wolfe reformulation of a mixed integer program partially convexifies a subset of the constraints, i.e., it implicitly adds all valid inequalities for the associated integer hull. Projecting an optimal basic solution of the reformulation’s LP relaxation to the original space does is in general not yield a basic solution of the original LP relaxation. Cutting planes in the original problem that are separated using a basis like Gomory mixed integer cuts are therefore not directly applicable. Range [22] (and others) proposed as a remedy to heuristically compute a basic solution and separate this auxiliary solution also with cutting planes that stem from a basis. This might not only cut off the auxiliary solution, but also the solution we originally wanted to separate. We discuss and extend Range’s ideas to enhance the separation procedure. In particular, we present alternative heuristics and consider additional valid inequalities strengthening the original LP relaxation before sepa-ration. Our full implementation, which is the first of its kind, is done within the GCG framework. We evaluated the effects on several problem classes. Our experiments show that the separated cuts strengthen the formulation on instances where the integrality gap is not too small. This leads to a reduced number of nodes and reduced solution times.