Korean Journal of Computational Design and Engineering (한국CDE학회논문집)

Society for Computational Design and Engineering (한국CDE학회)
권호 표지
DOI QR코드

DOI QR Code

A Method for RBF-based Approximate Optimization of Expensive Black Box Functions

고비용 블랙박스 함수의 RBF기반 근사 최적화 기법

  • Park, Sangkun (Dept. of Mechanical Engineering, Korean Nat'l Univ. of Transportation)
  • 박상근 (한국교통대학교 기계공학과)
  • Received : 2016.05.18
  • Accepted : 2016.06.21
  • Published : 2016.12.01
Download PDF
⟨ Previous Next ⟩

Abstract

This paper proposes a method for expensive black box optimization using radial basis functions (RBFs). The proposed algorithm is a computational strategy that uses a RBF model approximating the expensive black box function to predict an optimum. First, a RBF-based approximation technique is introduced and a sampling plan for estimation of the black box function is described. Then the proposed algorithm is explained, which presents the pseudo-codes for implementation and the detailed description of each step performed in the optimization process. In addition, numerical experiments will be given to analyze the performance of the proposed algorithm, by investigating computation accuracy, number of function evaluations, and convergence history. Finally, geometric distance problem as application example will be also presented for showing the algorithm applicability to different engineering problems.

Keywords

References

  1. Park, S., 2015, An Approximate Optimization Technique Using Radial Basis Functions, Transactions of the Korean National University of Transportation, 50 (scheduled for publication).
  2. Buhmann, M.D., 2003, Radial Basis Functions, Cambridge University Press.
  3. deBoor, C. and Ron, A., 1992, Computational Aspects of Polynomial Interpolation in Several Variable, Mathematics of Computation, 58, pp.705-727. https://doi.org/10.1090/S0025-5718-1992-1122061-0
  4. Lebensztajn, L., Marretto, C.A.R., Costa, M.C., and Coulomb, J.-L., 2004, Kriging: A Useful Tool for Electromagnetic Device Optimization, IEEE Trans Magn, 40, pp.1196-1199. https://doi.org/10.1109/TMAG.2004.824542
  5. Gutmann, H.M., 2001. A Radial Basis Function Method for Global Optimization, Journal of Global Optimization, 19(3), pp.201-227. https://doi.org/10.1023/A:1011255519438
  6. Clarke, S.M., Griebsch, J.H., and Simpson, T.W., 2005, Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses, Journal of Mechanical Design, 127, 1077-1087. https://doi.org/10.1115/1.1897403
  7. Forrester, A.I.J. and Keane, A.J., 2009, Recent Advances in Surrogate-based Optimization, Progress in Aerospace Sciences, 45, pp.50-79. https://doi.org/10.1016/j.paerosci.2008.11.001
  8. Bandler, J.W., Chen, Q.S., Dakroury, S.A., Mohamed, A.S., Bakr, M.H., Madsen, K., and Sondergaard, J., 2004, Space Mapping: The State of the Art, IEEE Trans on Microwave Theory and Techniques, 52(1), pp.337-361. https://doi.org/10.1109/TMTT.2003.820904
  9. Hemker, P.W. and Echeverria, D., 2007, A Trust-region Strategy for Manifold-mapping Optimization, Journal of Computational Physics, 224, pp.464-475. https://doi.org/10.1016/j.jcp.2007.04.003
  10. Alexandrov, N., Dennis, J.E., Lewis, R.M., and Torczon, V., 1998, A Trust Region Framework for Managing the Use of Approximation Models in Optimization, Structural Optimization, 15, pp.16-23. https://doi.org/10.1007/BF01197433
  11. Booker, A., Dennis, J., Frank, P., Serafini, D., Torczon, V., and Trosset, M., 1999, A Rigorous Framework for Optimization of Expensive Functions by Surrogates, Structural Optimization, 17, pp.1-13. https://doi.org/10.1007/BF01197708
  12. Madsen, K., Nielsen, H.B., and Tingleff, O., 2004, Methods for Non-Linear Least Squares Problems (2nd ed.), Informatics and Mathematical Modelling, Technical University of Denmark, DTU.
  13. Hedayat, A., Sloane, N., and Stufken, J., 1999, Orthogonal Arrays: Theory and Applications, Springer, Series in Statistics, Berlin: Springer.
  14. McKay, M., Conover, W., and Beckman, R., 1979, A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 21, pp.239-245.
  15. Wang, J.G. and Liu, G.R., 2002, On the Optimal Shape Parameters of Radial Basis Functions Used for 2D Meshless Methods, Computer Methods in Applied Mechanics and Engineering, 191, pp.2611-2630. https://doi.org/10.1016/S0045-7825(01)00419-4
  16. http://www.gamsworld.org/performance/selconglobal/selcongloballib.htm.
  17. Piegl, L. and Tiller, W., 1995, The NURBS Book, Springer-Verlag.