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Using QR Decomposition to Obtain a New Instance of Mesh Adaptive Direct Search with Uniformly Distributed Polling Directions

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Listed:
  • Benjamin Dyke

    (Washington State University)

  • Thomas J. Asaki

    (Washington State University)

Abstract

The purpose of this paper is to introduce a new instance of the Mesh Adaptive Direct Search (Mads) class of algorithms, which utilizes a more uniform distribution of poll directions than do other common instances, such as OrthoMads and LtMads. Our new implementation, called QrMads, bases its poll directions on an equal area partitioning of the n-dimensional unit sphere and the QR decomposition to obtain an orthogonal set of directions. While each instance produces directions which are dense in the limit, QrMads directions are more uniformly distributed in the unit sphere. This uniformity is the key to enhanced performance in higher dimensions and for constrained problems. The trade-off is that QrMads is no longer deterministic and at each iteration the set of polling directions is no longer orthogonal. Instead, at each iteration, the poll directions are only ‘nearly orthogonal,’ becoming increasingly closer to orthogonal as the mesh size decreases. Finally, we present a variety of test results on smooth, nonsmooth, unconstrained, and constrained problems and compare them to OrthoMads on the same set of problems.

Suggested Citation

  • Benjamin Dyke & Thomas J. Asaki, 2013. "Using QR Decomposition to Obtain a New Instance of Mesh Adaptive Direct Search with Uniformly Distributed Polling Directions," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 805-821, December.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:3:d:10.1007_s10957-013-0356-y
    DOI: 10.1007/s10957-013-0356-y
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    References listed on IDEAS

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    1. I. D. Coope & C. J. Price, 2000. "Frame Based Methods for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 261-274, November.
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    Cited by:

    1. Árpád Bűrmen & Jernej Olenšek & Tadej Tuma, 2015. "Mesh adaptive direct search with second directional derivative-based Hessian update," Computational Optimization and Applications, Springer, vol. 62(3), pages 693-715, December.
    2. Benjamin Van Dyke, 2014. "Equal Angle Distribution of Polling Directions in Direct-Search Methods," Journal of Optimization, Hindawi, vol. 2014, pages 1-15, July.
    3. Árpád Bűrmen & Iztok Fajfar, 2019. "Mesh adaptive direct search with simplicial Hessian update," Computational Optimization and Applications, Springer, vol. 74(3), pages 645-667, December.

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