zbMATH — the first resource for mathematics

Improved strategies for radial basis function methods for global optimization. (English) Zbl 1149.90120
The global optimization method based on the radial basis function model has been proposed by H.-M. Gutmann in 2001 [J. Glob. Optim. 19, No. 3, 201–227 (2001; Zbl 0972.90055)] for computationally expensive multimodal objective functions of modest dimensionality. In some cases the convergence of this method to the global minimum is slow. Two modifications proposed in this paper aim to enhance the efficienty of the original method. The first modification restricts the search region of the current iteration. In this way the authors want to ensure better balance between local and global search. The second modification defines a complete restart strategy when progress is slow during a preset number of iterations. Testing results are presented to illustrate the achieved improvements.

90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C59 Approximation methods and heuristics in mathematical programming
Full Text: DOI
[1] Björkman M., Holmström K. (2000). Global optimization of costly nonconvex functions using radial basis functions. Optim. Eng. 1(4):373–397 · Zbl 1035.90061
[2] Booker A.J., Dennis J.E., Frank P.D., Serafini D.B., Torczon V., Trosset M.W. (1999). A rigorous framework for optimization of expensive functions by surrogates. Struct. Optim. 17(1):1–13
[3] Buhmann M.D. (2003). Radial Basis Functions. Cambridge University Press, UK · Zbl 1038.41001
[4] Conn A.R., Scheinberg K., Toint Ph.L. (1997) Recent progress in unconstrained nonlinear optimization without derivatives. Math. Program. 79(3):397–414 · Zbl 0887.90154
[5] Dixon L.C.W., Szegö G. (1978). The global optimization problem: an introduction. In: Dixon L.C.W., Szegö G. (eds) Towards Global Optimization 2. North-Holland, Amsterdam, pp. 1–15
[6] Gomes C.P., Selman B., Crato N., Kautz H. (2000). Heavy-tailed phenomena in satisfiability and constraint satisfaction problems. J. Automated Reasoning 24:67–100 · Zbl 0967.68145
[7] Gutmann H.-M. (2001a). A radial basis function method for global optimization. J. Global Optim. 19(3):201–227 · Zbl 0972.90055
[8] Gutmann H.-M. (2001b). Radial basis function methods for global optimization. PhD Thesis, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK · Zbl 0972.90055
[9] Holmström K. (1999). The TOMLAB optimization environment in Matlab. Adv. Model. Optim. 1(1):47–69 · Zbl 1115.90404
[10] Horst R., Pardalos P.M., Thoai N.V. (2000). Introduction to Global Optimization, 2nd Edn. Kluwer, Dordrecht · Zbl 0966.90073
[11] Jones, D.R.: Global Optimization with Response Surfaces, presented at the Fifth SIAM Conference on Optimization, Victoria, Canada (1996).
[12] Jones D.R., Perttunen C.D., Stuckman B.E. (1993). Lipschitz optimization without the lipschitz constant. J. Optim. Theor. Appli. 78(1):157–181 · Zbl 0796.49032
[13] Jones D.R., Schonlau M., Welch W.J. (1998). Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4):455–492 · Zbl 0917.90270
[14] Marazzi M., Nocedal J. (2002). Wedge trust region methods for derivative free optimization. Math. Program. Ser. A 91:289–305 · Zbl 1049.90134
[15] 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 · Zbl 0415.62011
[16] Powell, M.J.D.: 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, pp. 105–210. Oxford University Press, (1992) · Zbl 0787.65005
[17] Powell M.J.D. (1996). A review of algorithms for thin plate spline interpolation in two dimensions. In: Fontanella F., Jetter K., Laurent P.J. (eds) Advanced Topics in Multivariate Approximation. World Scientific Publishing, River Edge, NJ, pp. 303–322 · Zbl 1273.65019
[18] Powell M.J.D. (1999). Recent research at cambridge on radial basis functions. In: Müller M., Buhmann M., Mache D. and Felten M. (eds) New Developments in Approximation Theory. International Series of Numerical Mathematics, Vol. 132, Birkhauser Verlag, Basel, pp. 215–232 · Zbl 0958.41501
[19] Powell M.J.D. (2000). UOBYQA: unconstrained optimization by quadratic approximation. Math. Program. 92:555–582 · Zbl 1014.65050
[20] Powell M.J.D. (2002). On trust region methods for unconstrained minimization without derivatives. Math. Program. 97:605–623 · Zbl 1106.90382
[21] Regis R.G., Shoemaker C.A. (2005). Constrained global optimization using radial basis functions. J. Global Optim. 31:153–171 · Zbl 1274.90511
[22] Rinnooy Kan A.H.G., Timmer G.T. (1987). Stochastic global optimization methods, part II: multi level methods. Math. Program. 39:57–78 · Zbl 0634.90067
[23] Serafini, D.B.: A framework for managing models in non-linear optimization of computationally expensive functions. Ph.D. Thesis, Department of Computational and Applied Mathematics, Rice (1998)
[24] Schoen F. (1993). A wide class of test functions for global optimization. J. Global Optim. 3:133–137 · Zbl 0772.90072
[25] The Mathworks Inc. Optimization Toolbox for use with MATLAB: User’s Guide, Version 3, Natick, MA (2004)
[26] Torczon V. (1997). On the convergence of pattern search algorithms. SIAM J. Optim. 7(1):1–25 · Zbl 0884.65053
[27] Torn A., Zilinskas A. (1989). Global Optimization. Lecture Notes in Computer Science, Vol. 350, Springer-Verlag, Berlin
[28] Ye K.Q., Li W., Sudjianto A. (2000). Algorithmic construction of optimal symmetric latin hypercube designs. J. Stat. Plan. Infer. 90:145–159 · Zbl 1109.62329
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.