Abstract
The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31-variable helicopter rotor blade design example and for a standard optimization test example.
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Barthelemy, J-F.M.; Haftka, R.T. 1993: Approximation concepts for optimum structural design — a review.Struct. Optim. 5, 129–144
Booker, A.J. 1994: DOE for computer output.Technical Report BCSTECH-94-052 Boeing Computer Services, Research and Technology, M/S 7L-68, Seattle, Washington 98124
Booker, A.J. 1996: Case studies in design and analysis of computer experiments. InProc. Section on Physical and Engineering Sciences, American Statistical Association
Booker, A.J.; Conn, A.R., Dennis, J.E. Jr.; Frank, P.D.; Trosset, M.W.; Torczon, V. 1995. Global modeling for optimization Boeing/IBM/Rice collaborative project 1995 final report.Technical Report ISSTECH-95-032, Boeing Information & Support Services, Research and Technology, M/S 7L-68, Seattle, Washington 98124
Booker, A.J.; Dennis, J.E., Jr.; Frank, P.D.; Serafini, D.B.; Torczon, V. 1997. Optimization using surrogate objectives on a helicopter test example.Technical Report SSGTECH-97-027, Roeing Shared Services Group, Applied Research & Technology, M/S 7L-68, Seattle, Washington 98124. Also available as Technical Report 97-31, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005-1892. To appear in: Borggaard, J.; Burns, J.; Cliff, E.; Schreck, S. (eds.)Optimal design and control. Cambridge, MA: Birkhauser
Burgee, S.L.; Giunta, A.A.; Balabanov, V.; Grossman, B.; Mason, W.H.; Narducci, R.; Haftka, R.T.; Watson, L.T. 1996: A coarse-grained parallel variable-complexity multidisciplinary optimization paradigm.Int. J. Supercomputing Appl. & High Performance Computing 10, 269–299
Conn, A.R.; Scheinberg, K.; Toint, Ph.L. 1997: On the convergence of derivative-free methods for unconstrained optimization. In: Iserles, A.; Buhmann, M. (eds.)Approximation theory and optimization: tributes to M.J.D. Powell, pp. 83–108. Cambridge: Cambridge University Press
Conn, A.R.; Toint, Ph.L. 1996: An algorithm using quadratic interpolation for unconstrained derivative free optimization. In: Di Pillo, G.; Giannessi, F. (eds.)Nonlinear optimization and applications, pp. 27–47. New York: Plenum Publishing
Cox, D.D.; John, S. 1997: SDO: a statistical method for global optimization. In: Alexandrov, N.; Hussaini, M.Y. (eds.)Multidisciplinary design optimization: state of the art, pp. 315–329. Philadelphia: SIAM
Currin, C.; Mitchell, T.; Morris, M.; Ylvisaker, D. 1988: A Bayesian approach to the design and analysis of computer experiments.Technical Report ORNL-6498, Oak Ridge National Laboratory
Davis, C. 1954: Theory of positive linear dependence.Amer. J. Math. 76, 733–746
De Boor, C.; Ron, A. 1992: Computational aspects of polynomial interpolation in several variables.Mathematics of Computation 58, 705–727
Dennis, J.E., Jr.; Torczon, V. 1991: Direct search methods on parallel machines.SIAM J. Optimiz. 1, 448–474
Dennis, J.E., Jr.; Torczon, V. 1997: Managing approximation models in optimization. In: Alexandrov, N.; Hussaini, M.Y. (eds.)Multidisciplinary design optimization: state-of-the-art, pp. 330–347. Philadelphia: SIAM. Also available as Technical Report 95-19, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005-1892
Dennis, J.E., Jr.; Walker, H.F. 1984: Inaccuracy in quasi-Newton methods: local improvement theorems.Math. Prog. Study 22, 70–85
Dixon, L.C.W.; Szegö G.P. (eds.) 1978:Towards global optimization 2. Amsterdam: North-Holland Pub. Co.
Efron, B.; Stein, C. 1981: The jackknife estimate of variance.Annals of Statistics 9, 586–596
Frank, P.D. 1995: Global modeling for optimization.SIAG/OPT Views-and-News 7, 9–12
Geisser, S.; Eddy, W.F. 1979: A predictive approach to model selection.J. Amer. Statistical Assoc. 74, 153–160
Giunta, A.A. 1997:Aircraft multidisciplinary optimization using design of experiments theory and response surface modeling methods. Ph.D. Thesis, Virginia Tech. Available as MAD 97-05-01, May 1997, Department of Aerospace and Ocean Engineering, Virginia Tech, 215 Randolph Hall, Blacksburg, Virginia 24061
Jones, D.R.; Schonlau, M.; Welch, W.J. 1997: A data analytic approach to Bayesian global optimization.Proc. ASA
Koehler, J.R.; Owen, A.B. 1996: Computer experiments. In: Ghosh, S.; Rao, C.R. (eds.)Handbook of statistics, volume 13, pp. 261–308. New York: Elsevier Science
Levine, D. 1996: Users guide to the PGAPack parallel genetic algorithm library.Technical Report ANL-95/18, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439 Available from URL ftp: //info.mcs.anl.gov/pub/pgapack/pgapack.tar.Z
Lewis, R.M.; Torczon, V. 1996: Pattern search algorithms for bound constrained minimization.Technical Report 96-20 ICASE, NASA Langley Research Center, Hampton, Virginia 23681-2199. To appear inSIAM J. Optimiz.
Lewis, R.M.; Torczon, V. 1996: Rank ordering and positive bases in pattern search algorithms.Technical Report 96-71, ICASE, NASA Langley Research Center, Hampton, Virginia 23681-2199. In revision forMath. Prog.
Lewis, R.M.; Torczon, V. 1997: Pattern search methods for linearly constrained minimization.Technical Report 98-03, ICASE, NASA Langley Research Center, Hampton, Virginia 23681-2199.SIAM J. Optimiz. (to appear)
McKay, M.D.; Conover, W.J.; Beckman, R.J. 1979 A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics 21, 239–245
Owen, A.B. 1992: Orthogonal arrays for computer experiments, integration and visualization.Statistica Sinica 2, 439–452
Rinnooy Kan, A.H.G.; Timmer, G.T. 1984: A stochastic approach to global optimization. In: Boggs, P.T.; Byrd, R.H.; Schnabel, R.B. (eds.)Numerical optimization 1984 (Proc. SIAM Conf. on Numerical Optimization), pp. 245–262. Philadelphia: SIAM
Ripley, B.D. 1988:Statistical inference for spatial processes. Cambridge: Cambridge University Press
Sacks, J.; Welch, W.J.; Mitchell, T.J.; Wynn, H.P. 1989: Design and analysis of computer experiments.Statistical Sci.4, 409–435
Serafini, D.B. 1998:A framework for managing models in nonlinear optimization of computationally expensive functions. Ph.D. Thesis, Rice University
Shultz, L.A.; Panda, B.; Tarzanin, F.J.; Derham, R.C.; Oh, B.K.; Dadone, L. 1994: Interdisciplinary analysis for advanced rotors-approach, capabilities and status. AHS Aeromechanics Specialists Conf., PS 4-1-4-15
Snir, M.; Otto S.W.; Huss-Lederman, S.; Walker, D.W.; Dongarra, J. 1996.MPI: The complete reference. Cambridge, MA: The MIT Press
Stein, M. 1987: Large sample properties of simulations using latin hypercube sampling.Technometrics 29, 143–151
Tang, B. 1993: Orthogonal array-based latin hypercubes.J. Amer. Statistical Assoc. 88, 1392–1397
Torczon, V. 1992: PDS: direct search methods for unconstrained optimization on either sequential or parallel machines.Technical Report 92-9, Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005-1892. In revision forACM Trans. Math. Software
Torczon, V. 1995: Pattern search methods for nonlinear optimization.SIAG/OPT Views & News 6, 7–11
Torczon, V. 1997: On the convergence of pattern search algorithms.SIAM J. Optimiz. 7, 1–25
Torczon, V.; Trosset, M.W. 1998: From evolutionary operation to parallel direct search: pattern search algorithms for numerical optimization.Computing Sci. & Statistics 29, 396–401
Trosset, M.W.; Torczon, V. 1997: Numerical optimization using computer experiments.Technical Report 97-38, ICASE, NASA Langley Research Center, Hampton, Virginia 23681-2199
Trosset, M.W. 1997: I know it when I see it: toward a definition of direct search methods.SIAG/OPT Views & News 9, 7–10
Watson, G.S. 1984: Smoothing and interpolation by kriging and with splines.Math. Geology 16, 601–615
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Booker, A.J., Dennis, J.E., Frank, P.D. et al. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization 17, 1–13 (1999). https://doi.org/10.1007/BF01197708
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DOI: https://doi.org/10.1007/BF01197708