Optimization and Engineering

, Volume 9, Issue 4, pp 311–339

An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization

  • Kenneth Holmström
  • Nils-Hassan Quttineh
  • Marcus M. Edvall
Article

Abstract

Response surface methods based on kriging and radial basis function (RBF) interpolation have been successfully applied to solve expensive, i.e. computationally costly, global black-box nonconvex optimization problems. In this paper we describe extensions of these methods to handle linear, nonlinear, and integer constraints. In particular, algorithms for standard RBF and the new adaptive RBF (ARBF) are described. Note, however, while the objective function may be expensive, we assume that any nonlinear constraints are either inexpensive or are incorporated into the objective function via penalty terms. Test results are presented on standard test problems, both nonconvex problems with linear and nonlinear constraints, and mixed-integer nonlinear problems (MINLP). Solvers in the TOMLAB Optimization Environment (http://tomopt.com/tomlab/) have been compared, specifically the three deterministic derivative-free solvers rbfSolve, ARBFMIP and EGO with three derivative-based mixed-integer nonlinear solvers, OQNLP, MINLPBB and MISQP, as well as the GENO solver implementing a stochastic genetic algorithm. Results show that the deterministic derivative-free methods compare well with the derivative-based ones, but the stochastic genetic algorithm solver is several orders of magnitude too slow for practical use. When the objective function for the test problems is costly to evaluate, the performance of the ARBF algorithm proves to be superior.

Keywords

Global optimization Radial basis functions Response surface model Surrogate model Expensive function CPU-intensive Optimization software Splines Mixed-integer nonlinear programming constraints 

Abbreviations

RBF

Radial basis function

CGO

Costly global optimization

MINLP

Mixed-integer nonlinear programming

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References

  1. Bakr MH, Bandler JW, Madsen K, Sondergaard J (2000) Review of the space mapping approach to engineering optimization and modeling. Optim Eng 1(3):241–276 MATHCrossRefMathSciNetGoogle Scholar
  2. Björkman M, Holmström K (2000) Global optimization of costly nonconvex functions using radial basis functions. Optim Eng 1(4):373–397 MATHCrossRefMathSciNetGoogle Scholar
  3. Gutmann H-M (1999) A radial basis function method for global optimization. Technical Report DAMTP 1999/NA22, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England Google Scholar
  4. Gutmann H-M (2001a) A radial basis function method for global optimization. J Glob Optim 19:201–227 MATHCrossRefMathSciNetGoogle Scholar
  5. Gutmann H-M (2001b) Radial basis function methods for global optimization. Doctoral thesis, Department of Numerical Analysis, Cambridge University, Cambridge, UK Google Scholar
  6. Holmström K (1999) The TOMLAB optimization environment in Matlab. Adv Model Optim 1(1):47–69 MATHGoogle Scholar
  7. Holmström K (2007) An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization. J Glob Optim. doi:10.1007/s10898-007-9256-8 MATHGoogle Scholar
  8. Holmström K, Edvall MM (2004) The TOMLAB optimization environment. In: Kallrath LGJ (ed) Modeling languages in mathematical optimization. Boston Google Scholar
  9. Jones DR (2002) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21:345–383 CrossRefGoogle Scholar
  10. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13:455–492 MATHCrossRefMathSciNetGoogle Scholar
  11. McKay M, Beckman R, Conover W (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–246 MATHCrossRefMathSciNetGoogle Scholar
  12. Powell MJD (1992) The theory of radial basis function approximation in 1990. In: Light W (ed) Advances in numerical analysis, vol. 2: wavelets, subdivision algorithms and radial basis functions. Oxford University Press, Oxford, pp 105–210 Google Scholar
  13. Powell MJD (1999) Recent research at Cambridge on radial basis functions. In: Buhmann MD, Felten M, Mache D, Müller MW (eds) New developments in approximation theory. Birkhäuser, Basel, pp 215–232 Google Scholar
  14. Regis RG, Shoemaker CA (2005) Constrained global optimization of expensive black box functions using radial basis functions. J Glob Optim 31(1):153–171 MATHCrossRefMathSciNetGoogle Scholar
  15. Regis RG, Shoemaker CA (2007) Improved strategies for radial basis function methods for global optimization. J Glob Optim 37(1):113–135 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Kenneth Holmström
    • 1
  • Nils-Hassan Quttineh
    • 1
  • Marcus M. Edvall
    • 2
  1. 1.Department of Mathematics and PhysicsMälardalen UniversityVästeråsSweden
  2. 2.Tomlab Optimization Inc.PullmanUSA

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