### Abstract

We introduce a Bayesian approach for obtaining the global optimum of multimodal functions. The set of observed minima of a multimodal function is viewed as a sample from a multinomial distribution, whose cells correspond to the basins of attraction of the local optima. We then derive the posterior distribution of the number of local optima. This posterior information is obtained from a Bayesian analysis and is used to construct a stopping criterion for a sequential random search method which finds the optimal tradeoff between reliability and computational effort. The computational complexity of this global optimization method is a strong function of the total number of local optima and a weak function of the dimensions of the configuration space. Application to four classical problems from the global optimization literature, a bifunctional catalytic reactor problem, and the conformation problem of Lennard-Jones microclusters is demonstrated. Comparisons with the Bayesian method of Boender and Rinooy Kan [Math. Program. 37, 59 (1987)] and the simulated annealing method of Dekkers and Aarts [Math. Program. 50, 367 (1991)] are provided and, in each case, the computational complexity of our method is shown to be smaller than that of these methods.

Original language | English (US) |
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Pages (from-to) | 6219-6232 |

Number of pages | 14 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 55 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

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## Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*55*(5), 6219-6232. https://doi.org/10.1103/PhysRevE.55.6219