Stochastic Optimization and the Gambler's Ruin Problem

Abstract An analogy between stochastic optimization and the gambler's ruin problem is used to estimate the expected value of the number of function evaluations required to reach the extremum of a special objective function with a pafrticular random walk. The objective function is the sum of the squares of the independent variables. The optimization is accomplished when the random walk enters a suitably defined small neighborhood of the extremum. The results indicate that for this objective function the expected number of function evaluations increases as the number of dimensions to the five halves power. Results of extensive computations of optimizing random walks in spaces with dimensions ranging from 2 to 30 agree with the analytically predicted behavior.

[1]  L. Haines.  The application of the annealing algorithm to the construction of exact optimal designs for linear-regression models , 1987 .

[2]  M. E. Johnson,et al.  Generalized simulated annealing for function optimization , 1986 .

[3]  D Ceperley,et al.  Quantum Monte Carlo , 1986, Science.

[4]  D. Mitra,et al.  Convergence and finite-time behavior of simulated annealing , 1986, Advances in Applied Probability.

[5]  G. Woodfin,et al.  An "Intelligent" Optical Design Program , 1984, Other Conferences.

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[7]  S. Geman,et al.  Diffusions for global optimizations , 1986 .

[8]  Donald Geman,et al.  Bayes Smoothing Algorithms for Segmentation of Binary Images Modeled by Markov Random Fields , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Debasis Mitra,et al.  A chaotic asynchronous algorithm for computing the fixed point of a nonnegative matrix of unit spectral radius , 1986, JACM.

[10]  Rammile Ettelaie,et al.  Residual entropy and simulated annealing , 1985 .

[11]  M. Waterman,et al.  Mapping DNA by stochastic relaxation , 1987 .

[12]  Kenneth J. Supowit,et al.  Simulated Annealing Without Rejected Moves , 1986, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[13]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[14]  F. Darema,et al.  Parallel Algorithms for Chip Placement by Simulated Annealing , 1987, IBM J. Res. Dev..

[15]  Alexander H. G. Rinnooy Kan,et al.  Stochastic methods for global optimization , 1984 .

[16]  Paul Molitor.  Layer assignment by simulated annealing , 1985 .

[17]  George Wolberg,et al.  Restoration of binary images using stochastic relaxation with annealing , 1985, Pattern Recognit. Lett..

[18]  P. Baldi.  Limit set of inhomogeneous Ornstein-Uhlenbeck processes, destabilization and annealing , 1986 .

[19]  Emile H. L. Aarts,et al.  Simulated Annealing: Theory and Applications , 1987, Mathematics and Its Applications.

[20]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[21]  B. Golden,et al.  Using simulated annealing to solve routing and location problems , 1986 .

[22]  A. Dress,et al.  Parsimonious phylogenetic trees in metric spaces and simulated annealing , 1987 .

[23]  Mark E. Johnson,et al.  Optimal deployment of missile interceptors , 1988 .

[24]  T. P. Cotter,et al.  An Attempt To Develop An "Intelligent" Lens Design Program , 1986, Other Conferences.