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A CARTOPT METHOD FOR BOUND-CONSTRAINED GLOBAL OPTIMIZATION

Published online by Cambridge University Press:  18 March 2014

B. L. ROBERTSON ,
C. J. PRICE  and
M. REALE
B. L. ROBERTSON*
Affiliation:
Department of Statistics, University of Wyoming, Laramie, Wyoming, USA
C. J. PRICE
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand email chrisj.price@canterbury.ac.nz email marco.reale@canterbury.ac.nz
M. REALE
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand email chrisj.price@canterbury.ac.nz email marco.reale@canterbury.ac.nz
*
✉brober25@uwyo.edu
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Abstract

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A stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

Keywords

CARTHalton sequencenumerical resultsrandom searchstochastic global optimization

MSC classification

Secondary: 90C26: Nonconvex programming, global optimization
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 2 , October 2013 , pp. 109 - 128
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Abramson, M. A., Audet, C., Dennis, J. E. and Le Digabel, S., “OrthoMADS: a deterministic MADS instance with orthogonal directions”, SIAM J. Optim. 20 (2008) 948–966 ; doi:10.1137/080716980.CrossRefGoogle Scholar
Ali, M. M., Khompatraporn, C. and Zabinsky, Z. B., “A numerical evaluation of several stochastic algorithms on selected continuous global optimization problems”, J. Global Optim. 31 (2005) 635–672 ; doi:10.1007/s10898-004-9972-2.CrossRefGoogle Scholar
Appel, M. J., LaBarre, R. and Radulovic, D., “On accelerated random search”, SIAM J. Optim. 14 (2003) 708–731 ; doi:10.1137/S105262340240063X.CrossRefGoogle Scholar
Audet, C. and Dennis, J. E. Jr, “Analysis of generalized pattern searches”, SIAM J. Optim. 13 (2003) 889–903 ; doi:10.1137/S1052623400378742.CrossRefGoogle Scholar
Brachetti, P., De Felice Ciccoli, M., Di Pillo, G. and Lucidi, S., “A new version of the Price’s algorithm for global optimization”, J. Global Optim. 10 (2010) 165–184 ; doi:10.1023/A:1008250020656.CrossRefGoogle Scholar
Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J., Classification and regression trees (Wadsworth International Group, Monterey, CA, 1984).Google Scholar
Dyer, M. E. and Frieze, A. M., “Computing the volume of convex bodies: a case where randomness provably helps”, Proc. Sympos. Appl. Math. 44 (1991) 123–169 ; doi:10.1090/psapm/044/1141926.CrossRefGoogle Scholar
Dyer, M., Frieze, A. and Kannan, R., “A random polynomial-time algorithm for approximating the volume of convex bodies”, J. ACM 38 (1991) 1–17 ; doi:10.1145/102782.102783.CrossRefGoogle Scholar
Ermakov, S. M., Zhigyavskii, A. A. and Kondratovich, M. V., “Comparison of some random search procedures for a global extremum”, USSR Comput. Math. Math. Phys. 29 (1989) 112–117 ; doi:10.1016/0041-5553(89)90054-2.CrossRefGoogle Scholar
Halton, J. H., “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals”, Numer. Math. 2 (1960) 84–90 ; doi:10.1007/BF01386213.CrossRefGoogle Scholar
Hirsch, M. J., Meneses, C. N., Pardalos, P. M. and Resende, M. G. C., “Global optimization by continuous GRASP”, Optim. Lett. 1 (2007) 201–212 ; doi:10.1007/s11590-006-0021-6.CrossRefGoogle Scholar
Horst, R. and Hoang, T., Global optimization: deterministic approaches, 3rd edn. (Springer, Berlin, 1996).CrossRefGoogle Scholar
Horst, R. and Pardalos, P. M., Handbook of global optimization (Kluwer, Dordrecht, 1995).CrossRefGoogle Scholar
Jones, D., Perttunen, C. D. and Stuckman, B. E., “Lipschitzian optimization without the Lipschitz constant”, J. Optim. Theory Appl. 79 (1993) 157–181 ; doi:10.1007/BF00941892.CrossRefGoogle Scholar
Martinez, W. L. and Martinez, A. R., Computational statistics handbook with Matlab (Chapman & Hall/CRC, Boca Raton, FL, 2002).Google Scholar
Moré, J. J., Garbow, B. S. and Hillstrom, K. E., “Testing unconstrained optimization software”, ACM Trans. Math. Softw. 7 (1981) 17–41 ; doi:10.1145/355934.355936.CrossRefGoogle Scholar
Price, C. J., Reale, M. and Robertson, B. L., “A cover partitioning method for bound constrained global optimization”, Optim. Meth. Softw. 27 (2012) 1059–1072 ; doi:10.1080/10556788.2011.557726.CrossRefGoogle Scholar
Price, W. L., “A controlled random search procedure for global optimisation”, Comput. J. 4 (1977) 367–370 ; doi:10.1093/comjnl/20.4.367.CrossRefGoogle Scholar
Rinnooy Kan, A. H. G. and Timmer, G. T., “Stochastic global optimization methods part II: Multi level methods”, Math. Program. 39 (1987) 57–78 ; doi:10.1007/BF02592071.CrossRefGoogle Scholar
Robertson, B. L., “Direct search methods for nonsmooth problems using global optimization techniques”, Ph. D. Thesis, University of Canterbury, Christchurch, New Zealand, 2010.Google Scholar
Robertson, B. L., Price, C. J. and Reale, M., “CARTopt: a random search method for nonsmooth unconstrained optimization”, Comput. Optim. Appl. 56 (2013) 291–315 ; doi:10.1007/s10589-013-9560-9.CrossRefGoogle Scholar
Srivastava, A., Han, E.-H., Kumar, V. and Singh, V., “Parallel formulations of decision-tree classification algorithms”, High Performance Data Mining 3 (2002) 237–261 ; doi:10.1007/0-306-47011-X_2.CrossRefGoogle Scholar
Yıldız, O. T. and Dikmen, O., “Parallel univariate decision trees”, Pattern Recog. Lett. 28 (2007) 825–832 ; doi:10.1016/j.patrec.2006.11.009.CrossRefGoogle Scholar
Zabinsky, Z. B., Stochastic adaptive search for global optimization (Kluwer, Dordrecht, 2003).CrossRefGoogle Scholar
Zabinsky, Z. B. et al. , “Random search algorithms”, in: Wiley encyclopedia of operations research and management science (ed. Cochran, J. J.), (Wiley, Hoboken, NJ, 2010) ; doi:10.1002/9780470400531.Google Scholar
Zhigljavsky, A. A., Theory of global random search (Kluwer, Dordrecht, 1991).CrossRefGoogle Scholar