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Towards an Accurate Solution of Wireless Network Design Problems

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Combinatorial Optimization (ISCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

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Abstract

The optimal design of wireless networks has been widely studied in the literature and many optimization models have been proposed over the years. However, most models directly include the signal-to-interference ratios representing service coverage conditions. This leads to mixed-integer linear programs with constraint matrices containing tiny coefficients that vary widely in their order of magnitude. These formulations are known to be challenging even for state-of-the-art solvers: the standard numerical precision supported by these solvers is usually not sufficient to reliably guarantee feasible solutions. Service coverage errors are thus commonly present. Though these numerical issues are known and become evident even for small-sized instances, just a very limited number of papers has tried to tackle them, by mainly investigating alternative non-compact formulations in which the sources of numerical instabilities are eliminated. In this work, we explore a new approach by investigating how recent advances in exact solution algorithms for linear and mixed-integer programs over the rational numbers can be applied to analyze and tackle the numerical difficulties arising in wireless network design models.

This work was partly conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (BMBF, Grant no. 05M14ZAM). It was also partially supported by the Einstein Center for Mathematics Berlin through Project ROUAN (MI4) (ROUAN) and by BMBF through Project VINO (Grant no. 05M13ZAC) and Project ROBUKOM (Grant no. 05M10ZAA) [4].

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Notes

  1. 1.

    This is the definition of feasibility used by the academic MINLP solver SCIP  [1, 26]. While we do not know for certain the numerical definitions used by closed-source commercial solvers, we think that they follow a similar practice.

  2. 2.

    For more details on WiMAX networks, see [9].

  3. 3.

    The smallest MIP has 808 variables, 900 constraints, and 8 000 nonzeros, the largest instance contains 32 436 variables, 33 300 constraints, and 1 231 200 nonzeros.

  4. 4.

    Although with this kind of unreliability, this does not matter anymore, note that the numerical difficulties during the solving process are also reflected in the lower objective values obtained by the unscaled models.

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Authors and Affiliations

  1. Department of Mathematical Optimization, Zuse Institute Berlin (ZIB), Takustr. 7, 14195, Berlin, Germany

    Fabio D’Andreagiovanni & Ambros M. Gleixner

  2. DFG Research Center MATHEON, Einstein Center for Mathematics (ECMath), StraĂźe des 17. Juni 136, 10623, Berlin, Germany

    Fabio D’Andreagiovanni

  3. Institute for System Analysis and Computer Science, National Research Council of Italy (IASI-CNR), via dei Taurini 19, 00185, Rome, Italy

    Fabio D’Andreagiovanni

Authors
  1. Fabio D’Andreagiovanni
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  2. Ambros M. Gleixner
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Correspondence to Fabio D’Andreagiovanni .

Editor information

Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

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D’Andreagiovanni, F., Gleixner, A.M. (2016). Towards an Accurate Solution of Wireless Network Design Problems. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_12

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  • DOI: https://doi.org/10.1007/978-3-319-45587-7_12

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  • Publisher Name: Springer, Cham

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