Sigrún Andradóttir
Abstract
This article is concerned with proving the almost sure and global convergence of a broad class of algorithms for solving simulation optimization problems with countably infinite number of feasible points. We first describe the class of simulation optimization algorithms under consideration and discuss how the estimate of the optimal solution should be chosen when the feasible region of the underlying optimization problem is countably infinite. Then, we present a general result that guarantees the global convergence with probability one of the simulation optimization algorithms in this class. The assumptions of this result are sufficiently weak to allow the algorithms under consideration to be efficient, in that they are not required to either allocate the same amount of computer effort to all the feasible points these algorithms visit, or to spend an increasing amount of computer effort per iteration as the number of iterations grows. This article concludes with a discussion of how our assumptions can be satisfied and also generalized.
- Agrawal, R. 1995. Sample mean based index policies with o(log n) regret for the multi-armed bandit problem. Adv. Appl. Prob. 27, 1054--1078.Google Scholar
- Alrefaei, M. H. and Andradóttir, S. 1999. A simulated annealing algorithm with constant temperature for discrete stochastic optimization. Manage. Sci. 45, 748--764. Google Scholar
- Alrefaei, M. H. and Andradóttir, S. 2001. A modification of the stochastic ruler method for discrete stochastic optimization. Europ. J. Operat. Res. 133, 1, 160--182.Google Scholar
- Alrefaei, M. H. and Andradóttir, S. 2005. Discrete stochastic optimization using variants of the stochastic ruler method. Naval Res. Log. 52, 344--360.Google Scholar
- Andradóttir, S. 1995. A method for discrete stochastic optimization. Management Science 41, 1946--1961. Google Scholar
- Andradóttir, S. 1996. A global search method for discrete stochastic optimization. SIAM J. Optim. 6, 2 (May), 513--530.Google Scholar
- Andradóttir, S. 1998. Simulation optimization. In Handbook of Simulation: Principles, Methodology, Advances, Applications, and Practice, J. Banks, Ed. Wiley, New York.Google Scholar
- Andradóttir, S. 1999. Accelerating the convergence of random search methods for discrete stochastic optimization. ACM Trans. Model. Comput. Simulat. 9, 4, 349--380. Google Scholar
- Andradóttir, S. 2000. Rate of convergence of random search methods for discrete optimization using steady-state simulation. Tech. Rep., Georgia Institute of Technology.Google Scholar
- Dembo, A. and Zeitouni, O. 1993. Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.Google Scholar
- Devroye, L. 1976. Probabilistic search as a strategy selection procedure. IEEE Trans. Syst., Man, Cyber. 6, 315--321.Google Scholar
- Durrett, R. 1991. Probability: Theory and Examples. Duxbury Press, Belmont, CA.Google Scholar
- Fox, B. L. and Heine, G. W. 1996. Probabilistic search with overrides. Ann. Appl. Probab. 6, 1087--1094.Google Scholar
- Fu, M. C. 2002. Optimization for simulation: Theory vs. practice. INFORMS J. Comput. 14, 3, 192--215. Google Scholar
- Gelfand, S. B. and Mitter, S. K. 1989. Simulated annealing with noisy or imprecise energy measurements. J. Optim. Theory Appl. 62, 49--62.Google Scholar
- Gong, W.-B., Ho, Y.-C., and Zhai, W. 1999. Stochastic comparison algorithm for discrete optimization with estimation. SIAM J. Optim. 10, 2, 384--404. Google Scholar
- Grimmett, G. R. and D. R. Stirzaker. 1992. Probability and Random Processes, 2nd ed. Oxford University Press, Oxford, UK.Google Scholar
- Gutjahr, W. J. and Pflug, G. Ch. 1996. Simulated annealing for noisy cost functions. J. Global Optim. 8, 1--13.Google Scholar
- Ho, Y. C., Sreenivas, R. S., and Vakili, P. 1992. Ordinal optimization of DEDS. J. Disc. Event Dynam. Syst. 2, 61--88.Google Scholar
- Homem-de-Mello, T. 2003. Variable-sample methods for stochastic optimization. ACM Trans. Model. Comput. Simul. 13, 2 (4), 108--133. Google Scholar
- Hong, L. J. and Nelson, B. L. 2006. Discrete optimization via simulation using COMPASS. Operat. Res. 54, 115--129. Google Scholar
- Kim, S.-H. and Nelson, B. L. 2001. A fully sequential procedure for indifference-zone selection in simulation. ACM Trans. Model. Comput. Simul. 11, 251--273. Google Scholar
- Kleywegt, A. J., Shapiro, A., and Homem-de-Mello, T. 2001. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12, 2, 479--502. Google Scholar
- Lai, T. L. and Robbins, H. 1985. Asymptotically efficient adaptive allocation rules. Adv. Appl. Math. 6, 4--22.Google Scholar
- Louhichi, S. 2003. Moment inequalities for sums of certain dependent random variables. Theory Prob. Appl. 47, 4 (Dec.), 649--664.Google Scholar
- Nelson, B. L., Swann, J., Goldsman, D., and Song, W. 2001. Simple procedures for selecting the best simulated system when the number of alternatives is large. Oper. Res. 49, 950--963. Google Scholar
- Norkin, V. I., Pflug, G. Ch., and Ruszczyński, A. 1998. A branch and bound method for stochastic global optimization. Math. Prog. A 83, 425--450. Google Scholar
- Pichitlamken, J. and Nelson, B. L. 2003. A combined procedure for optimization via simulation. ACM Trans. Model. Comput. Simul. 13, 2, 155--179. Google Scholar
- Prudius, A. A. and Andradóttir, S. 2006. Balanced explorative and exploitative search with estimation. Submitted for publication.Google Scholar
- Révész, P. 1968. The Laws of Large Numbers. Academic Press, New York.Google Scholar
- Shi, L. and Ólafsson, S. 2000. Nested partitions method for stochastic optimization. Method. Comput. Appl. Prob. 2, 3, 271--291.Google Scholar
- Stout, W. F. 1974. Almost Sure Convergence. Academic Press, New York.Google Scholar
- Yakowitz, S. and Lugosi, E. 1990. Random search in the presence of noise, with applications to machine learning. SIAM J. Sci. Stat. Comput. 11, 702--712. Google Scholar
- Yakowitz, S. and Lowe, W. 1991. Nonparametric bandit methods. Ann. Operat. Res. 28, 297--312. Google Scholar
- Yan, D. and Mukai, H. 1992. Stochastic discrete optimization. SIAM Journal on Control and Optimization 30, 3, 594--612. Google Scholar
Index Terms
- Simulation optimization with countably infinite feasible regions: Efficiency and convergence
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