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The Feasibility Pump (FP) is one of the best-known primal heuristics for mixed-integer programming (MIP): more than 15 papers suggested various modifications of all of its steps. So far, no variant considered information across multiple iterations, but all instead maintained the principle to optimize towards a single reference integer point. In this paper, we evaluate the usage of multiple reference vectors in all stages of the FP algorithm. In particular, we use LP-feasible vectors obtained during the main loop to tighten the variable domains before entering the computationally expensive enumeration stage. Moreover, we consider multiple integer reference vectors to explore further optimizing directions and introduce alternative objective scaling terms to balance the contributions of the distance functions and the original MIP objective. Our computational experiments demonstrate that the new method can improve performance on general MIP test sets. In detail, our modifications provide a 29.3% solution quality improvement and 4.0% running time improvement in an embedded setting, needing 16.0% fewer iterations over a large test set of MIP instances. In addition, the method’s success rate increases considerably within the first few iterations. In a standalone setting, we also observe a moderate performance improvement, which makes our version of FP suitable for the two main use-cases of the algorithm.
Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a $4\%$ decrease in solve time, and an $8\%$ decrease in number of nodes over affected instances of MIPLIB 2017.