Abstract
Hide-and-Seek is a powerful yet simple and easily implemented continuous simulated annealing algorithm for finding the maximum of a continuous function over an arbitrary closed, bounded and full-dimensional body. The function may be nondifferentiable and the feasible region may be nonconvex or even disconnected. The algorithm begins with any feasible interior point. In each iteration it generates a candidate successor point by generating a uniformly distributed point along a direction chosen at random from the current iteration point. In contrast to the discrete case, a single step of this algorithm may generateany point in the feasible region as a candidate point. The candidate point is then accepted as the next iteration point according to the Metropolis criterion parametrized by anadaptive cooling schedule. Again in contrast to discrete simulated annealing, the sequence of iteration points converges in probability to a global optimum regardless of how rapidly the temperatures converge to zero. Empirical comparisons with other algorithms suggest competitive performance by Hide-and-Seek.
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References
Aarts, E. H. L. and J. H. M. Korst (1988),Simulated Annealing and Boltzmann Machines, Wiley, Chichester.
Aarts, E. H. L. and P. J. M. van Laarhoven (1989), Simulated annealing: an introduction,Statistica Neerlandica 43, 31–52.
Aluffi-Pentini, F., V. Parisi, and F. Zirilli (1985), Global optimization and stochastic differential equations,Journal of Optimization Theory and Applications 47, 1–16.
Ballard, D. H., C. O. Jelinek, and R. Schinzinger (1974), An algorithm for the solution of constrained generalized polynomial programming problems,The Computer Journal 17, 261–266.
Bélisle, C. J. P. (1992), Convergence theorems for a class of simulated annealing algorithms on ℝd,Journal of Applied Probability 29, 885–895.
Bélisle, C. J. P., H. E. Romeijn, and R. L. Smith (1993), Hit-and-run algorithms for generating multivariate distributions,Mathematics of Operations Research 18, 255–266.
Berbee, H. C. P., C. G. E. Boender, A. H. G. Rinnooy Kan, C. L. Scheffer, R. L. Smith, and J. Telgen (1987), Hit-and-run algorithms for the identification of nonredundant linear inequalities,Mathematical Programming 37, 184–207.
Boender, C. G. E., R. J. Caron, A. H. G. Rinnooy Kan, J. F. McDonald, H. E. Romeijn, R. L. Smith, J. Teigen, and A. C. F. Vorst (1991), Shake-and-Bake algorithms for generating uniform points on the boundary of bounded polyhedra,Operations Research 39, 945–954.
Bohachevsky, I. O., M. E. Johnson, and M. L. Stein (1986), Generalized simulated annealing for function optimization,Technometrics 28, 209–217.
Bremmerman, H. (1970), A method of unconstrained global optimization,Mathematical Biosciences 9, 1–15.
Corana, A., M. Marchesi, C. Martini, and S. Ridella (1987), Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithm,ACM Transactions on Mathematical Software 13, 262–280.
De Haan, L. F. M. (1981), Estimation of the minimum of a function using order statistics,Journal of the American Statistical Association 76, 467–469.
Dekker, A. and E. H. L. Aarts (1991), Global optimization and simulated annealing,Mathematical Programming 50, 367–393.
Dixon, L. C. W. and G. P. Szegö (1978),Towards Global Optimization 2, North-Holland, Amsterdam, The Netherlands.
Doob, J. L. (1953),Stochastic Processes, Wiley, New York, NY.
Hajek, B. (1988), Cooling schedules for optimal annealing,Mathematics of Operations Research 13, 311–329.
Hwang, C. (1980), Laplace's method revisited: weak convergence of probability measures,The Annals of Probability 8, 1177–1182.
Kaufman, D. E. and R. L. Smith (1991), Optimal direction choice for Hit-and-Run sampling, Technical Report 90-08, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, Michigan.
Kirkpartick, S., C. D. Gelatt Jr., and M. P. Vecchi (1983), Optimization by simulated annealing,Science 20, 671–680.
Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller (1953), Equations of state calculations by fast computing machines,The Journal of Chemical Physics 21, 1087–1092.
Nummelin, E. (1984),General Irreducible Markov Chains and Non-Negative Operators, Cambridge University Press, Cambridge, U.K.
Patel, N. R., R. L. Smith, and Z. B. Zabinsky (1988), Pure adaptive search in Monte Carlo optimization,Mathematical Programming 43, 317–328.
Pncus, M. (1968), A closed form solution for certain programming problems,Operations Research 16, 690–694.
Pincus, M. (1970), A Monte-Carlo method for the approximate solution of certain types of constrained optimization problems,Operations Research 18, 1225–1228.
Rinnooy Kan, A. H. G. and G. T. Timmer (1984), Stochastic methods for global optimization,American Journal of Mathematical and Management Sciences 4, 7–40.
Rinnooy Kan, A. H. G. and G. T. Timmer (1987a), Stochastic global optimization methods; part I: clustering methods,Mathematical Programming 39, 27–56.
Rinnooy Kan, A. H. G. and G. T. Timmer (1987b), Stochastic global optimization methods; part II: multilevel methods,Mathematical Programming 39, 57–78.
Romeijn, H.E. and R. L. Smith (1990), Sampling through random walks. Technical Report 90-02. Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI.
Rubinstein, R. Y. (1981),Simulation and the Monte Carlo Method, Wiley, New York, NY.
Smith, R. L. (1984), Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions,Operations Research 32, 1296–1308.
Timmer, G. T. (1984),Global optimization: a stochastic approach. PhD thesis, Erasmus University Rotterdam, Rotterdam, The Netherlands.
Zabinsky, Z. B. and R. L. Smith (1992), Pure adaptive search in global optimization,Mathematical Programming 53, 323–338.
Zabinsky, Z. B., R. L. Smith, J. F. McDonald, H. E. Romeijn, and D. E. Kaufman (1993), Improving Hit-and-Run for global optimization,Journal of Global Optimization 3, 171–192.
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This material is based on work supported by a NATO Collaborative Research Grant, no. 0119/89.
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Romeijn, H.E., Smith, R.L. Simulated annealing for constrained global optimization. J Glob Optim 5, 101–126 (1994). https://doi.org/10.1007/BF01100688
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DOI: https://doi.org/10.1007/BF01100688