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A Linear Constrained Optimization Benchmark for Probabilistic Search Algorithms: The Rotated Klee-Minty Problem

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Theory and Practice of Natural Computing (TPNC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11324))

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Abstract

The development, assessment, and comparison of randomized search algorithms heavily rely on benchmarking. Regarding the domain of constrained optimization, the small number of currently available benchmark environments bears no relation to the vast number of distinct problem features. The present paper advances a proposal of a scalable linear constrained optimization problem that is suitable for benchmarking Evolutionary Algorithms. By comparing two recent Evolutionary Algorithm variants, the linear benchmarking environment is demonstrated.

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Notes

  1. 1.

    The code related to the BBOB-constrained suite under development is available in the development branch on http://github.com/numbbo/coco/development.

  2. 2.

    We omit use of the redundant constraints introduced in [2]. That is, the parameter \(\varvec{h}\) in [2] is considered to be an array of all-zeros for our Klee-Minty representation.

  3. 3.

    The Klee-Minty problem usually assumes the non-negativity of the parameter vector components. The upper bound \(\hat{\varvec{y}}\) is set in accordance with the translation in Eq. (6).

  4. 4.

    In the field of EA, several methods to treat box-constraints do exist, e.g. by random reinitialization inside the box or by repair of violated components.

  5. 5.

    The choice of the translation \(\varvec{t}\) represents an empirically motivated compromise between complexity and numerical stability for a wide range of search space dimensions.

  6. 6.

    This is a first suggestion; larger search space dimensions can easily be included.

  7. 7.

    All three termination criteria were considered for both EA variants to realize the experimental results displayed in Sect. 4. Instead, Random Search (RS) omits the third termination criterion as stagnations are likely.

  8. 8.

    It is recommended to use the mentioned constraint violation definition. As there exist multiple different ways to define the constraint violation, algorithm developers may use their definition of choice. However, a detailed explanation is obligatory to ensure the comparability of algorithm test results.

  9. 9.

    For performance improvements, the \(\epsilon \) threshold is initially set to zero in all algorithm runs, i.e. the \(\epsilon \)-level ordering is replaced with the lexicographic ordering (10).

  10. 10.

    A more detailed explanation of the ECDF construction and interpretation is provided in [3]. Notice, this version of the rotated Klee-Minty benchmark omits the use of the bootstrapping approach mentioned in that respective paper.

  11. 11.

    https://github.com/hellwigm/RotatedKleeMintyProblem.

References

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Acknowledgements

This work was supported by the Austrian Science Fund (FWF) under grant P29651-N32. The article is also based upon work from COST Action CA15140 supported by COST (European Cooperation in Science and Technology).

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Authors and Affiliations

  1. Vorarlberg University of Applied Sciences, Research Centre PPE, Campus V, Hochschulstraße 1, 6850, Dornbirn, Austria

    Michael Hellwig & Hans-Georg Beyer

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  1. Michael Hellwig
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  2. Hans-Georg Beyer
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Correspondence to Michael Hellwig .

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Editors and Affiliations

  1. School of Business, University College Dublin, Dublin, Ireland

    David Fagan

  2. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  3. School of Business, University College Dublin, Dublin, Ireland

    Michael O'Neill

  4. Escuela Politécnica, University of Extremadura, Caceres, Spain

    Miguel A. Vega-Rodríguez

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Hellwig, M., Beyer, HG. (2018). A Linear Constrained Optimization Benchmark for Probabilistic Search Algorithms: The Rotated Klee-Minty Problem. In: Fagan, D., Martín-Vide, C., O'Neill, M., Vega-Rodríguez, M.A. (eds) Theory and Practice of Natural Computing. TPNC 2018. Lecture Notes in Computer Science(), vol 11324. Springer, Cham. https://doi.org/10.1007/978-3-030-04070-3_11

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  • DOI: https://doi.org/10.1007/978-3-030-04070-3_11

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