Abstract
The goal of this paper is to organize some of the mathematical and algorithmic aspects of the space-mapping technique for continuous optimization with expensive function evaluations. First, we consider the mapping from the fine space to the coarse space when the models are vector-valued functions and when the space-mapping (nonlinear) least-squares residual is nonzero. We show how the sensitivities of the space mapping can be used to deal with space-mapping surrogates of the fine model. We derive a framework where it is possible to design globally convergent trust-region methods to minimize such fine-model surrogates.
We consider also a different perspective of space mapping and apply it, for sake of simplicity, to the situation where the models are scalar functions. The space mapping is defined in a way where it is reasonable to assume that it is point-to-point. We prove that the surrogate model built by composition of the space mapping and the coarse model is a regular function. We also discuss trust-region methods in this context.
Similar content being viewed by others
References
N. M. Alexandrov, J. E. Dennis, R. M. Lewis, and V. Torczon, “A trust region framework for managing the use of approximation models in optimization,” Structural Optimization, vol. 15, pp. 16–23, 1998.
M. H. Bakr, J. W. Bandler, R. M. Biernacki, S. H. Chen, and K. Madsen, “A trust region agressive space mapping algorithm for EM optimization,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 2412–2425, 1998.
M. H. Bakr, J. W. Bandler, K. Madsen, and J. Søndergaard, “Review of the space mapping approach to engineering optimization and modeling,” Optimization and Engineering, vol. 1, pp. 241–276, 2000.
M. H. Bakr, J. W. Bandler, K. Madsen, and J. Søndergaard, “An introduction to the space mapping technique,” Optimization and Engineering, vol. 2, pp. 369–384, 2002.
J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Space mapping technique for electromagnetic optimization,” IEEE Trans. Microwave Theory Tech., vol. 42, pp. 2536–2544, 1994.
J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, and K. Madsen, “Electromagnetic optimization exploiting agressive space mapping,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2874–2882, 1995.
R. G. Carter, “On the global convergence of trust region algorithms using inexact gradient information,” SIAM J. Numer. Anal., vol. 28, pp. 251–265, 1991.
A. R. Conn, N. I. M. Gould, and P. L. Toint, “Trust-region methods,” MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2000.
J. E. Dennis, “Surrogate modelling and space mapping for engineering optimization,” A summary of the Danish Technical University November 2000 Workshop, Technical Report TR00–35, Department of Computational and Applied Mathematics, Rice University, 2000.
J. E. Dennis, S.-B. B. Li, and R. A. Tapia, A unified approach to global convergence of trust region methods for nonsmooth optimization,” Math. Programming, vol. 68, pp. 319–346, 1995.
S. Leary, A. Bhaskar, and A. Keane, “A constraint mapping approach to the structural optimization of an expensive model using surrogates,” in Surrogate Modelling and Space Mapping for Engineering Optimization, H. B. Nielsen (Ed.), DK-2800, Lyngby–Denmark, 2000, Department of Mathematical Modelling, Technical University of Denmark.
J. J. Moré, “Recent developments in algorithms and software for trust regions methods,” in Mathematical Programming. The State of Art, A. Bachem, M. Grotschel, and B. Korte (Eds.), Springer Verlag: New York, 1983, pp. 258–287.
H. B. Nielsen (Ed.), “Surrogate modelling and space mapping for engineering optimization,” DK-2800, Lyngby–Denmark, 2000, Department of Mathematical Modelling, Technical University of Denmark.
M. J. D. Powell, “Convergence properties of a class of minimization algorithms,” in Nonlinear Programming 2, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson (Eds.), Academic Press: New York, 1975, pp. 1–27.
J. Søndergaard, “Non-linear optimization using space mapping,” Master's Thesis, Department of Mathematical Modelling, Technical University of Denmark, 1999.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vicente, L.N. Space Mapping: Models, Sensitivities, and Trust-Regions Methods. Optimization and Engineering 4, 159–175 (2003). https://doi.org/10.1023/A:1023968629245
Issue Date:
DOI: https://doi.org/10.1023/A:1023968629245