Convergence of simulated annealing by the generalized transition probability

TL;DR

This paper establishes the weak ergodicity of a generalized transition probability in simulated annealing, providing a mathematical basis for convergence to the global minimum, supported by an explicit one-dimensional example.

Contribution

It proves weak ergodicity for a generalized transition probability in simulated annealing, extending convergence results beyond traditional methods.

Findings

Weak ergodicity of the inhomogeneous Markov process is proven.

Explicit example shows fast convergence to the optimal value.

Analyticity requirements ensure convergence in the solvable model.

Abstract

We prove weak ergodicity of the inhomogeneous Markov process generated by the generalized transition probability of Tsallis and Stariolo under power-law decay of the temperature. We thus have a mathematical foundation to conjecture convergence of simulated annealing processes with the generalized transition probability to the minimum of the cost function. An explicitly solvable example in one dimension is analyzed in which the generalized transition probability leads to a fast convergence of the cost function to the optimal value. We also investigate how far our arguments depend upon the specific form of the generalized transition probability proposed by Tsallis and Stariolo. It is shown that a few requirements on analyticity of the transition probability are sufficient to assure fast convergence in the case of the solvable model in one dimension.