**Scale Invariance Properties in the Simulated Annealing Algorithm** |

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**Authors:** | Fleischer M A Jacobson S H |

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**Institution:** | (1) Institute for Systems Research, University of Maryland, College Park;(2) Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, USA |

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**Abstract:** | The Boltzmann distribution used in the steady-state analysis of the simulated annealing algorithm gives rise to several scale invariant properties. Scale invariance is first presented in the context of parallel independent processors and then extended to an abstract form based on lumping states together to form new aggregate states. These lumped or aggregate states possess all of the mathematical characteristics, forms and relationships of states (solutions) in the original problem in both first and second moments. These scale invariance properties therefore permit new ways of relating objective function values, conditional expectation values, stationary probabilities, rates of change of stationary probabilities and conditional variances. Such properties therefore provide potential applications in analysis, statistical inference and optimization. Directions for future research that take advantage of scale invariance are also discussed. |

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**Keywords:** | scale invariance self-similarity simulated annealing Markov chains branch and probability bound nested partition algorithms |

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