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RENS

The optimal rounding

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Abstract

This article introduces rens, the relaxation enforced neighborhood search, a large neighborhood search algorithm for mixed integer nonlinear programs (MINLPs). It uses a sub-MINLP to explore the set of feasible roundings of an optimal solution \(\bar{x}\) of a linear or nonlinear relaxation. The sub-MINLP is constructed by fixing integer variables \(x_j\) with \(\bar{x} _{j} \in \mathbb {Z}\) and bounding the remaining integer variables to \(x_{j} \in \{ \lfloor \bar{x} _{j} \rfloor , \lceil \bar{x} _{j} \rceil \}\). We describe two different applications of rens: as a standalone algorithm to compute an optimal rounding of the given starting solution and as a primal heuristic inside a complete MINLP solver. We use the former to compare different kinds of relaxations and the impact of cutting planes on the so-called roundability of the corresponding optimal solutions. We further utilize rens to analyze the performance of three rounding heuristics implemented in the branch-cut-and-price framework scip. Finally, we study the impact of rens when it is applied as a primal heuristic inside scip. All experiments were performed on three publicly available test sets of mixed integer linear programs (MIPs), mixed integer quadratically constrained programs (MIQCPs), and MINLP s, using solely software which is available in source code. It turns out that for these problem classes 60 to 70 % of the instances have roundable relaxation optima and that the success rate of rens does not depend on the percentage of fractional variables. Last but not least, rens applied as primal heuristic complements nicely with existing primal heuristics in scip.

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Notes

  1. Stage 3 of the Feasibility Pump solves (a reformulation of) the original MIP with a new objective function. It minimizes the distance to an infeasible point gained from the pumping algorithm; more precisely to the one which was closest to the polyhedron associated to the LP relaxation. For details, see [9].

  2. This holds, to a certain extent, for all general MINLP test sets that the author is aware of.

  3. For a detailed discussion of the shifted geometric mean, see Achterberg [2, Appendix A3].

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Acknowledgments

Many thanks go to Ambros M. Gleixner and Daniel E. Steffy for their thorough proof-reading and to two anonymous reviewers for their helpful comments. This research has been supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.

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

    Timo Berthold

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Berthold, T. RENS. Math. Prog. Comp. 6, 33–54 (2014). https://doi.org/10.1007/s12532-013-0060-9

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