Abstract
In the present work, we explore a general framework for the design of new minimization algorithms with desirable characteristics, namely, supervisor-searcher cooperation. We propose a class of algorithms within this framework and examine a gradient algorithm in the class. Global convergence is established for the deterministic case in the absence of noise and the convergence rate is studied. Both theoretical analysis and numerical tests show that the algorithm is efficient for the deterministic case. Furthermore, the fact that there is no line search procedure incorporated in the algorithm seems to strengthen its robustness so that it tackles effectively test problems with stronger stochastic noises. The numerical results for both deterministic and stochastic test problems illustrate the appealing attributes of the algorithm.
References
Dennis, J. E., and Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
Fletcher, R., Practical Methods of Optimization, 2nd Edition, John Wiley and Sons, Chichester, England, 1987.
Grippo, L., Lampariello, F., and Lucidi, S., A Nonmonotone Line Search Technique for Newton's Methods, SIAM Journal on Numerical Analysis, Vol. 23, pp. 707–716, 1986.
Kiefer, J., and Wolfowitz, J., Stochastic Estimation of the Maximum of a Regression Function, Annals of Mathematical Statistics, Vol. 23, pp. 462–466, 1952.
Robbins, H., and Monro, S., A Stochastic Approximation Method, Annals of Mathematical Statistics, Vol. 22, pp. 400–407, 1951.
Benveniste, M., Metivier, M., and Priouret, P., Adaptive Algorithms and Stochastic Approximation, Springer Verlag, Berlin, Germany, 1990.
Kushner, H. J., and Clark, D. C., Stochastic Approximation Methods for Constrained and Unconstrained Systems, Springer, New York, NY, 1978.
Kushner, H. J., and Yin, G. G., Stochastic Approximation Algorithms and Applications, Springer, New York, NY, 1997.
Ljung, L., System Identification Theory for the User, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.
Uryas'ev, S. P., A Stochastic Quasigradient Algorithm with Variable Metric, Annals of Operations Research, Vol. 39, pp. 251–267, 1992.
Elster, C., and Neumaier, A., A Trust-Region Method for the Optimization of Noisy Functions, Computing, Vol. 58, pp. 31–46, 1997.
Yan, D., and Mukai, H., Optimization Algorithm with Probabilistic Estimation, Journal of Optimization Theory and Applications, Vol. 79, pp. 345–371, 1993.
Conn, A. R., Scheinberg, K., and Toint, P. L., Recent Progress in Unconstrained Nonlinear Optimization without Derivatives, Mathematical Programming, Vol. 79, pp. 397–414, 1997.
Powell, M. J. D., Direct Search Algorithms for Optimization Calculations, Acta Numerica, pp. 287–336, 1998.
Anderson, E. J., and Ferris, M. C., A Direct Search Algorithms for Optimization with Noisy Function Evaluations, Working Paper 97-010, Australian Graduate School of Management, 1997.
Barton, R. R., and Ivey, J. S., Nelder-Mead Simplex Modification for Simulation Optimization, Management Science, Vol. 42, pp. 954–972, 1996.
Fu, M. C., Optimization via Simulation: A Review, Annals of Operations Research, Vol. 53, pp. 199–247, 1994.
Barzilai, J., and Borwein, M., Two-Point Stepsize Gradient Methods, IMA Journal of Numerical Analysis, Vol. 8, pp. 141–148, 1988.
Raydan, M., On the BB Choice of Steplength for the Gradient Method, IMA Journal of Numerical Analysis, Vol. 13, pp. 321–326, 1993.
Raydan, M., The Barzilai and Borwein Gradient Method for the Large-Scale Unconstrained Minimization Problem, SIAM Journal on Optimization, Vol. 7, pp. 26–33., 1997.
Dai Y. H., and Liao L. Z., R-Linear Convergence of the Barzilai and Borwein Gradient Method, Report ICM–99-039, Institute of Computational Mathematics and Scientific Computing, Beijing, China, 1999.
MorÉ, J., Garbow, B. S., and Hillstrom, K. E., Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software, Vol. 7, pp. 17–41, 1981.
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Liu, W., Dai, Y.H. Minimization Algorithms Based on Supervisor and Searcher Cooperation. Journal of Optimization Theory and Applications 111, 359–379 (2001). https://doi.org/10.1023/A:1011986402461
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DOI: https://doi.org/10.1023/A:1011986402461