Abstract
In this paper, we present a stochastic optimization algorithm based on the idea of the gradient method which incorporates a new adaptive-precision technique. Because of this new technique, unlike recent methods, the proposed algorithm adaptively selects the precision without any need for prior knowledge on the speed of convergence of the generated sequence. With this new technique, the algorithm can avoid increasing the estimation precision unnecessarily, yet it retains its favorable convergence properties. In fact, it tries to maintain a nice balance between the requirements for computational accuracy and those for computational expediency. Furthermore, we present two types of convergence results delineating under what assumptions what kinds of convergence can be obtained for the proposed algorithm.
Similar content being viewed by others
References
Rubinstein, R. Y.,Monte Carlo Optimization Simulation and Sensitivity of Queueing Networks, John Wiley and Sons, New York, New York, 1986.
Cao, X. R.,Sensitivity Estimates Based on One Realization of a Stochastic System, Journal of Statistical Computational Simulation, Vol. 27, pp. 211–232, 1987.
Dvoretsky, A.,On Stochastic Approximation, Proceedings of the 3rd Berkeley Symposium of Mathematical Statistics and Probability, Vol. 1, pp. 39–55, 1956.
Glynn, P. W.,Optimization of Stochastic Systems, Proceedings of the SCS Winter Simulation Conference, San Diego, California, pp. 52–59, 1986.
Kiefer, J., andWolfowitz, J.,Stochastic Estimation of the Maximum of a Regression Function, Annals of Mathematical Statistics, Vol. 23, pp. 452–466, 1952.
Robbins, H., andMonro, S.,Stochastic Approximation Methods, Annals of Mathematical Statistics, Vol. 22, pp. 400–407, 1951.
Kushner, H. J.,Stochastic Approximation Algorithms for the Local Optimization of Functions with Nonunique Stationary Points, IEEE Transactions on Automatic Control, Vol. 17, pp. 646–654, 1974.
Kushner, H. J., andGavin, T.,Stochastic Approximation Type Methods for Constrained Systems: Algorithm and Numerical Results, IEEE Transactions on Automatic Control, Vol. 19, pp. 349–357, 1974.
Maheshwari, S., andMukai, H.,An Optimization Algorithm Driven by Probabilistic Simulation, Proceedings of the 25th IEEE Conference on Decision and Control, Athens, Greece, pp. 1703–1705, 1986.
Wardi, Y.,A Stochastic Algorithm for Optimization Problems with Continua of Inequalities, Journal of Optimization Theory and Applications, Vol. 56, pp. 285–311, 1988.
Wardi, Y.,A Stochastic Steepest-Descent Algorithm, Journal of Optimization Theory and Applications, Vol. 59, pp. 307–323, 1988.
Klessig, R., andPolak, E.,An Adaptive-Precision Gradient Method for Optimal Control, SIAM Journal on Control and Optimization, Vol. 11, pp. 80–93, 1973.
Mukai, H., andPolak, E.,On the Use of Approximations in Algorithms for Optimization Problems with Equality and Inequality Constraints, SIAM Journal on Numerical Analysis, Vol. 15, pp. 674–693, 1978.
Armijo, L.,Minimization of Functions Having Continuous Partial Derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1–3, 1966.
Polak, E.,Computational Methods in Optimization: A Unified Approach, Academic Press, New York, New York, 1971.
Polak, E., Sargent, R. W. H., andSebastian, D. J.,On the Convergence of Sequential Minimization Algorithms, Journal of Optimization Theory and Applications, Vol. 14, pp. 439–442, 1974.
Ho, Y. C., andCao, X.,Perturbation Analysis and Optimization of Queueing Networks, Journal of Optimization Theory and Applications, Vol. 4, 559–582, 1983.
Yan, D.,Methods for Stochastic Optimization, Washington University, D.Sc. Thesis, 1990.
Yan, D., andMukai, H.,An Optimization Algorithm with Probabilistic Estimation, Report No. 92-10, Center for Optimization and Semantic Control, Department of Systems Science and Mathematics, Washington University, St. Louis, Missouri, 1992.
Author information
Authors and Affiliations
Additional information
Communicated by D. Q. Mayne
The work reported here was supported in part by NSF Grant No. ECS-85-06249 and USAF Grant No. AFOSR-89-0518. The authors wish to thank the anonymous reviewers whose careful reading and criticism have helped them improve the paper considerably.
Rights and permissions
About this article
Cite this article
Yan, D., Mukai, H. Optimization algorithm with probabilistic estimation. J Optim Theory Appl 79, 345–371 (1993). https://doi.org/10.1007/BF00940585
Issue Date:
DOI: https://doi.org/10.1007/BF00940585