Direct search algorithms for optimization calculations | Acta Numerica | Cambridge Core (original) (raw)
Abstract
Many different procedures have been proposed for optimization calculations when first derivatives are not available. Further, several researchers have contributed to the subject, including some who wish to prove convergence theorems, and some who wish to make any reduction in the least calculated value of the objective function. There is not even a key idea that can be used as a foundation of a review, except for the problem itself, which is the adjustment of variables so that a function becomes least, where each value of the function is returned by a subroutine for each trial vector of variables. Therefore the paper is a collection of essays on particular strategies and algorithms, in order to consider the advantages, limitations and theory of several techniques. The subjects addressed are line search methods, the restriction of vectors of variables to discrete grids, the use of geometric simplices, conjugate direction procedures, trust region algorithms that form linear or quadratic approximations to the objective function, and simulated annealing. We study the main features of the methods themselves, instead of providing a catalogue of references to published work, because an understanding of these features may be very helpful to future research.
Information
Type
Research Article
Copyright
Copyright © Cambridge University Press 1998
Access options
Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)
Article purchase
Temporarily unavailable
References
Brent, R. P. (1973), Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Brodlie, K. W. (1975), ‘A new direction set method for unconstrained minimization without evaluating derivatives’, J. Inst. Math. Appl. 15, 385–396.CrossRefGoogle Scholar
Conn, A. R., Scheinberg, K. and Toint, Ph. L. (1997a), ‘On the convergence of derivative free methods for unconstrained optimization’, in Approximation Theory and Optimization (Buhmann, M. D. and Iserles, A., eds), Cambridge University Press, Cambridge, pp. 83–108.Google Scholar
Conn, A. R., Scheinberg, K. and Toint, Ph. L. (1997b), ‘Recent progress in unconstrained nonlinear optimization without derivatives’, Math. Prog. 79, 397–414.CrossRefGoogle Scholar
Dennis, J. E. and Torczon, V. (1991), ‘Direct search methods on parallel machines’, SIAM J. Optim. 1, 448–474.CrossRefGoogle Scholar
Elster, C. and Neumaier, A. (1995), ‘A grid algorithm for bound constrained optimization of noisy functions’, IMA J. Numer. Anal. 15, 585–608.Google Scholar
Fletcher, R. (1987), Practical Methods of Optimization, Wiley, Chichester.Google Scholar
Gill, P. E., Murray, W. and Wright, M. H. (1981), Practical Optimization, Academic Press, London.Google Scholar
Goldberg, D. E. (1989), Genetic Algorithms in Search, Optimizationand Machine Learning, Addison-Wesley, Reading, MA.Google Scholar
Grippo, L., Lampariello, F. and Lucidi, S. (1988), ‘Global convergence and stabilization of unconstrained minimization methods without derivatives’, J. Optim. Theory Appl. 56, 385–406.CrossRefGoogle Scholar
Hooke, R. and Jeeves, T. A. (1961), ‘Direct search solution of numerical and statistical problems’, J. Assoc. Comput. Mach. 8, 212–229.CrossRefGoogle Scholar
Kelley, C. T. (1997), ‘Detection and remediation of stagnation in the Nelder–Mead algorithm using a sufficient decrease condition’, North Carolina State University Report CRSC-TR97-2.Google Scholar
van Laarhoven, P. J. M. and Aarts, E. H. L. (1987), Simulated Annealing: Theory and Applications, Reidel, Dordrecht.CrossRefGoogle Scholar
Lucidi, S. and Sciandrone, M. (1997), ‘On the global convergence of derivative free methods for unconstrained optimization’, preprint, Università di Roma ‘La Sapienza’, Italy.Google Scholar
McKinnon, K. I. M. (1997), ‘Convergence of the Nelder–Mead simplex method to a nonstationary point’, preprint (to be published in SIAM J. Optim.).CrossRefGoogle Scholar
Nelder, J. A. and Mead, R. (1965), ‘A simplex method for function minimization’, Comput. J. 7, 308–313.CrossRefGoogle Scholar
Powell, M. J. D. (1964), ‘An efficient method for finding the minimum of a function of several variables without calculating derivatives’, Comput. J. 7, 155–162.Google Scholar
Powell, M. J. D. (1973), ‘On search directions for minimization algorithms’, Math. Prog. 4, 193–201.CrossRefGoogle Scholar
Powell, M. J. D. (1994), ‘A direct search optimization method that models the objective and constraint functions by linear interpolation’, in Advances in Optimization and Numerical Analysis (Gomez, S. and Hennart, J-P., eds), Kluwer Academic, Dordrecht, pp. 51–67.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1986), Numerical Recipes: The Art of Scientific Computation, Cambridge University Press, Cambridge.Google Scholar
Rosenbrock, H. H. (1960), ‘An automatic method for finding the greatest or least value of a function’, Comput. J. 3, 175–184.CrossRefGoogle Scholar
Smith, C. S., (1962), ‘The automatic computation of maximum likelihood estimates’, N.C.B. Sci. Dept. Report SC 846/MR/40.Google Scholar
Spendley, W., Hext, G. R. and Himsworth, F. R. (1962), ‘Sequential application of simplex designs in optimisation and evolutionary operation’, Technometrics 4, 441–461.Google Scholar
Toint, Ph. L. and Callier, F. M. (1977), ‘On the accelerating property of an algorithm for function minimization without calculating derivatives’, J. Optim. Theory Appl. 23, 531–547.CrossRefGoogle Scholar
Torczon, V. (1997), ‘On the convergence of pattern search algorithms’, SIAM J. Optim. 7, 1–25.CrossRefGoogle Scholar
Winfield, D. (1973), ‘Function minimization by interpolation in a data table’, J. Inst. Math. Appl. 12, 339–347.CrossRefGoogle Scholar
Wright, M. H. (1996), ‘Direct search methods: once scorned, now respectable’, in Numerical Analysis 1995 (Griffiths, D. F. and Watson, G. A., eds), Addison Wesley Longman, Harlow, pp. 191–208.Google Scholar