Suppose a sequential sample is taken from an unknown discrete probability distribution on an unknown range of integers, in an effort to sample its maximum. A crucial issue is an appropriate stopping rude determining when to terminate the sampling process. We approach this problem from a Bayesian perspective, and derive stopping rules that minimize loss functions which assign a loss to the sample size and to the deviation between the maximum in the sample and the true (unknown) maximum. We will show that our rules offer an extremely simple approximate solution to the well-known problem to terminate the Multistart method for continuous global optimization.
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Betro, B. and F. Schoen (1987), Sequential Stopping Rules for the Multistart Algorithm in Global Optimization, Mathematical Programming 38.
Boender, C. G. E. and A. H. G. Rinnooy Kan (1987), Bayesian Stopping Rules for Multi-start Global Optimization Methods, Mathematical Programming 36.
Boender C. G. E. and A. H. G. Rinnooy Kan (1990), A Bayesian Learning Procedure for the (s, Q) Inventory Policy, Statistica Neerlandica 44, 3.
De Groot M. H. (1978), Optimal Statistical Decisions, McGraw-Hill, New-York.
Dixon L. C. W. and G. P. Szegö (1975), Towards Global Optimisation, North-Holland, Amsterdam.
Lin, S. (1965), Computer Solutions for the Travelling Salesman Problem, Bell. Syst. Tech. J. 44.
Lindley D. V. (1978), Bayesian Statistics: A Review, Society for Industrial and Applied Mathematics, Philadelphia.
Piccioni, M. and A. Ramponi (1991), Stopping Rules for the Multistart Method When Different Local Minima Have Different Function Values, to appear in Optimization.
Press H. P., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling (1986), Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New-York.
Rinnooy Kan, A. H. G. and G. T. Timmer (1987), Stochastic Global Optimization Methods. Part I: Clustering Methods, Part II; Multi Level Methods, Mathematical Programming 39.
Wilks S. S. (1962), Mathematical Statistics, Wiley, New-York.
Zielinski, R. (1981), A Statistical Estimate of the Structure of Multi-Extremal Problems, Mathematical Programming 21.
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Boender, C.G.E., Rinnooy Kan, A.H.G. On when to stop sampling for the maximum. J Glob Optim 1, 331–340 (1991). https://doi.org/10.1007/BF00130829
- Bayesian stopping rules