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Undercover: a primal MINLP heuristic exploring a largest sub-MIP

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Abstract

We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover, such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP.

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Notes

  1. Alternatively, we could recompute the reference solution to obtain values within the current bounds.

  2. If we want to apply Undercover more aggressively, we can try to recover from infeasibility by reordering the fixing sequence, e.g., such that the variable for which the fixing failed will be the first one in the reordered sequence. This is a simple version of a restarting mechanism. Restarting techniques are commonly used in solving SAT problems [35].

  3. If we want to apply Undercover aggressively and allow for solving the covering problem multiple times, the following two strategies can be used. First, during the fix-and-propagate routine, variables outside the precomputed cover may be fixed simultaneously. In this case, the fixing of some of the yet unfixed variables in the cover might become redundant. Recomputing the cover with \(\alpha _i = 1\) for all \(i\) with local bounds \(\hat{L}_i = \hat{U}_i\) may yield a smaller number of remaining variable fixings. Second, if no feasible fixings for the cover variables in \(\mathcal C \) are found, we can solve the covering problem again with an additional cutoff constraint \(\sum _{i\in \mathcal C } (1 - \alpha _i) + \sum _{i\not \in \mathcal C } \alpha _i \ge 1\) and try once more. However, both techniques appear to be computationally too expensive for the standard setting that we explored in our computational experiments.

  4. The source code is publicly available within SCIP  2.1.1 and can be found at [47].

  5. With a stall node limit, we terminate if no improving solutions are found within a certain number of branch-and-bound nodes after the discovery of the current incumbent.

  6. For one instance, the feasibility status had not been decided within 500 nodes.

  7. A primal cutoff is an upper bound on the objective function that results in branch-and-bound nodes with worse dual bound not being explored.

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Acknowledgments

Many thanks to Tobias Achterberg, J. Christopher Beck, Christina Burt, Angela Glover, Stefan Vigerske, the associate editor, and two anonymous reviewers for their valuable comments. This research has been supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin, http://www.matheon.de.

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  1. Department of Optimization, Zuse Institute Berlin, Takustr 7, 14195 , Berlin, Germany

    Timo Berthold & Ambros M. Gleixner

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  1. Timo Berthold
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Correspondence to Timo Berthold.

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Berthold, T., Gleixner, A.M. Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Math. Program. 144, 315–346 (2014). https://doi.org/10.1007/s10107-013-0635-2

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