Advertisement

Journal of Optimization Theory and Applications

, Volume 159, Issue 3, pp 805–821 | Cite as

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

  • Benjamin Van Dyke
  • Thomas J. Asaki
Article

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.

Keywords

Mesh adaptive direct search (Mads) algorithms Derivative-free optimization 

References

  1. 1.
    Audet, C., Dennis, J. Jr.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(1), 188–217 (2006) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abramson, M.A., Audet, C.: Convergence of mesh adaptive direct search to second-order stationary points. SIAM J. Optim. 17(2), 606–619 (2006) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Clarke, F.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1983) MATHGoogle Scholar
  4. 4.
    Vicente, L., Custódio, A.: Analysis of direct searches for discontinuous functions. Math. Program. 133(1–2), 299–325 (2012) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Rockafellar, R.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32(2), 257–280 (1980) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Torczon, V.: On the convergence of pattern search algorithms. SIAM J. Optim. 7(1), 1–25 (1997) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Abramson, M., Audet, C., Dennis, J. Jr., Le Digabel, S.: ORTHOMADS: a deterministic MADS instance with orthogonal directions. SIAM J. Optim. 10(2), 948–966 (2009) CrossRefGoogle Scholar
  8. 8.
    Coope, I.D., Price, C.J.: Frame-based methods for unconstrained optimization. J. Optim. Theory Appl. 107(2), 261–274 (2000) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2(1), 84–90 (1960) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Leopardi, P.: A partition of the unit sphere into regions of equal area and small diameter. Electron. Trans. Numer. Anal. 25, 309–327 (2006) MathSciNetMATHGoogle Scholar
  11. 11.
    Leopardi, P.: Distributing points on the sphere: partitions, separation, quadrature and energy. Ph.D. thesis, University of New South Wales (2007) Google Scholar
  12. 12.
    Leopardi, P.: Diameter bounds for equal area partitions of the unit sphere. Electron. Trans. Numer. Anal. 35, 1–16 (2009) MathSciNetMATHGoogle Scholar
  13. 13.
    Audet, C., Custódio, A.L., Dennis, J. Jr.: Erratum: mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 18(4), 1501–1503 (2008) CrossRefGoogle Scholar
  14. 14.
    Moré, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7(1), 17–41 (1981) CrossRefMATHGoogle Scholar
  15. 15.
    Lukšan, L., Vlček, J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Tech. Rep. No. 798, Institute of Computer Science, Academy of Sciences of the Czech Republic (2000) Google Scholar
  16. 16.
    Karmitsa, N.: Test problems for large-scale nonsmooth minimization. Tech. Rep. No. B. 4/2007, University of Jyväskylä (2007) Google Scholar
  17. 17.
    Ledoux, M.: The Concentration of Measure Phenomenon. Am. Math. Soc., Providence (2001) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsWashington State UniversityPullmanUSA

Personalised recommendations