Abstract
We present the AQUARS (A QUAsi-multistart Response Surface) framework for finding the global minimum of a computationally expensive black-box function subject to bound constraints. In a traditional multistart approach, the local search method is blind to the trajectories of the previous local searches. Hence, the algorithm might find the same local minima even if the searches are initiated from points that are far apart. In contrast, AQUARS is a novel approach that locates the promising local minima of the objective function by performing local searches near the local minima of a response surface (RS) model of the objective function. It ignores neighborhoods of fully explored local minima of the RS model and it bounces between the best partially explored local minimum and the least explored local minimum of the RS model. We implement two AQUARS algorithms that use a radial basis function model and compare them with alternative global optimization methods on an 8-dimensional watershed model calibration problem and on 18 test problems. The alternatives include EGO, GLOBALm, MLMSRBF (Regis and Shoemaker in INFORMS J Comput 19(4):497–509, 2007), CGRBF-Restart (Regis and Shoemaker in J Global Optim 37(1):113–135 2007), and multi level single linkage (MLSL) coupled with two types of local solvers: SQP and Mesh Adaptive Direct Search (MADS) combined with kriging. The results show that the AQUARS methods generally use fewer function evaluations to identify the global minimum or to reach a target value compared to the alternatives. In particular, they are much better than EGO and MLSL coupled to MADS with kriging on the watershed calibration problem and on 15 of the test problems.
Similar content being viewed by others
References
Abramson, M.A.: NOMADm Version 4.6 User’s Guide. Unpublished manuscript (2007)
Abramson M.A., Audet C.: Convergence of mesh adaptive direct search to second-order stationary points. SIAM J. Optim. 17(2), 606–619 (2006)
Aleman D.M., Romeijn H.E., Dempsey J.F.: A response surface approach to beam orientation optimization in intensity modulated radiation therapy treatment planning. INFORMS J. Comput. 21, 62–76 (2009)
Audet C., Dennis J.E. Jr.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(2), 188–217 (2006)
Björkman M., Holmström K.: Global optimization of costly nonconvex functions using radial basis functions. Optim. Eng. 1, 373–397 (2000)
Boender C.G.E., Rinnooy Kan A.H.G., Timmer G.T., Stougie L.: A stochastic method for global optimization. Math. Program. 22, 125–140 (1982)
Booker A.J., Dennis J.E., Frank P.D., Serafini D.B., Torczon V., Trosset M.W.: A rigorous framework for optimization of expensive functions by surrogates. Struct. Optim. 17(1), 1–13 (1999)
Buhmann M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003)
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization (2000)
Conn A.R., Scheinberg K., Vicente L.N.: Introduction to Derivative-Free Optimization. SIAM, Philadelphia (2009)
Cressie N.: Statistics for Spatial Data. Wiley, New York (1993)
Csendes T.: Nonlinear parameter estimation by global optimization—efficiency and reliability. Acta Cybern. 8, 361–370 (1988)
Dixon L.C.W., Szegö G.: The global optimization problem: an introduction. In: Dixon, L.C.W., Szegö, G. (eds) Towards Global Optimization, vol. 2, pp. 1–15. North-Holland, Amsterdam (1978)
Egea J.A., Vazquez E., Banga J.R., Marti R.: Improved scatter search for the global optimization of computationally expensive dynamic models. J. Global Optim. 43(2–3), 175–190 (2009)
Forrester A.I.J., Sobester A., Keane A.J.: Engineering Design via Surrogate Modelling a Practical Guide. Wiley, New York (2008)
Giunta A.A., Balabanov V., Haim D., Grossman B., Mason W.H., Watson L.T., Haftka R.T.: Aircraft multidisciplinary design optimisation using design of experiments theory and response surface modelling. Aeronaut. J. 101(1008), 347–356 (1997)
Glover, F.: A template for scatter search and path relinking. In: Hao, J.-K., Lutton, E., Ronald, E., Schoenauer, M., Snyers, D. (eds.) Artificial Evolution, Lecture Notes in Computer Science, vol. 1363, pp. 13–54. Springer, Berlin (1998)
Gutmann H.-M.: A radial basis function method for global optimization. J. Global Optim. 19(3), 201–227 (2001)
Holmström K.: An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization. J. Global Optim. 41(3), 447–464 (2008)
Huang D., Allen T.T., Notz W.I., Zeng N.: Global optimization of stochastic black-box systems via sequential kriging meta-models. J. Global Optim. 34(3), 441–466 (2006)
Jakobsson S., Patriksson M., Rudholm J., Wojciechowski A.: A method for simulation based optimization using radial basis functions. Optim. Eng. 11(4), 501–532 (2009)
Jones D.R., Schonlau M., Welch W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)
Kleijnen J.P.C.: Design and Analysis of Simulation Experiments. Springer, Berlin (2008)
Kleijnen J.P.C.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192(3), 707–716 (2009)
Kolda T.G., Lewis R.M., Torczon V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45(3), 385–482 (2003)
Laguna M., Marti R.: Scatter Search: Methodology and Implementations in C. Kluwer, Boston (2003)
Lasdon L., Duarte A., Glover F., Laguna M., Marti R.: Adaptive memory programming for constrained global optimization. Comput. Oper. Res. 37(8), 1500–1509 (2010)
Lophaven, S.N., Nielsen, H.B., Søndergaard, J.: DACE: A Matlab Kriging Toolbox, Version 2.0. Technical Report IMM-TR-2002-12, Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby(2002)
Marsden A.L., Wang M., Dennis J.E. Jr., Moin P.: Optimal aeroacoustic shape design using the surrogate management framework. Optim. Eng. 5(2), 235–262 (2004)
Myers R.H., Montgomery D.C.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley, New York (1995)
Nocedal J., Wright S.J.: Numerical Optimization. Springer, New York (1999)
Oeuvray R., Bierlaire M.: BOOSTERS: a derivative-free algorithm based on radial basis functions. Int. J. Model. Simul. 29(1), 26–36 (2009)
Oeuvray, R.: Trust-Region Methods Based on Radial Basis Functions With Application To Biomedical Imaging. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne (2005)
Powell M.J.D.: The theory of radial basis function approximation in 1990. In: Light, W. (eds) Advances in Numerical Analysis: Wavelets, Subdivision Algorithms and Radial Basis Functions, vol. 2, pp. 105–210. Oxford University Press, Oxford (1992)
Powell M.J.D.: UOBYQA: unconstrained optimization by quadratic approximation. Math. Program. 92, 555–582 (2002)
Powell M.J.D.: The NEWUOA software for unconstrained optimization without derivatives. In: Di Pillo, G., Roma, M. (eds) Large-Scale Nonlinear Optimization, pp. 255–297. Springer, USA (2006)
Regis R.G., Shoemaker C.A.: A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J. Comput. 19(4), 497–509 (2007)
Regis R.G., Shoemaker C.A.: Improved strategies for radial basis function methods for global optimization. J. Global Optim. 37(1), 113–135 (2007)
Rinnooy Kan A.H.G., Timmer G.T.: Stochastic global optimization methods, part II: multi level methods. Math. Program. 39, 57–78 (1987)
Sacks J., Welch W.J., Mitchell T.J., Wynn H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989)
Schoen F.: A wide class of test functions for global optimization. J. Global Optim. 3, 133–137 (1993)
Sendin J.O.H, Banga J.R., Csendes T.: Extensions of a multistart clustering algorithm for constrained global optimization problems. Ind. Eng. Chem. Res. 48(6), 3014–3023 (2009)
Shoemaker C.A., Regis R.G., Fleming R.C.: Watershed calibration using multistart local optimization and evolutionary optimization with radial basis function approximation. Hydrol. Sci. J. 52(3), 450–465 (2007)
Simpson T.W., Mauery T.M., Korte J.J., Mistree F.: Kriging metamodels for global approximation in simulation-based multidisciplinary design optimization. AIAA J. 39(12), 2233–2241 (2001)
The Mathworks, Inc.: Matlab Optimization Toolbox: User’s Guide, Version 4. Natick, MA (2009)
Ugray Z., Lasdon L., Plummer J., Glover F., Kelley J., Marti R.: Scatter search and local NLP solvers: a multistart framework for global optimization. INFORMS J. Comput. 19(3), 328–340 (2007)
Viana, F.A.C.: SURROGATES Toolbox User’s Guide, Version 2.1, http://sites.google.com/site/felipeacviana/surrogatestoolbox (2010)
Villemonteix J., Vazquez E., Walter E.: An informational approach to the global optimization of expensive-to-evaluate functions. J. Global Optim. 44(4), 509–534 (2009)
Ye K.Q., Li W., Sudjianto A.: Algorithmic construction of optimal symmetric latin hypercube designs. J. Stat. Plan. Inference 90, 145–159 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Regis, R.G., Shoemaker, C.A. A quasi-multistart framework for global optimization of expensive functions using response surface models. J Glob Optim 56, 1719–1753 (2013). https://doi.org/10.1007/s10898-012-9940-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9940-1