Convergence of an annealing algorithm.

*(English)*Zbl 0581.90061The annealing algorithm is a stochastic optimization method which has attracted attention because of its success with certain difficult problems, including NP-hard combinatorial problems such as the travelling salesman, Steiner trees and others. There is an appealing physical analogy for its operation, but a more formal model seems desirable. In this paper we present such a model and prove that the algorithm converges with probability arbitrarily close to 1. We also show that there are cases where convergence takes exponentially long - that is, it is no better than a deterministic method. We study how the convergence rate is affected by the form of the problem. Finally we describe a version of the algorithm that terminates in polynomial time and allows a good deal of ’practical’ confidence in the solution.

##### Keywords:

global optimization; metropolis method; hill climbing; local improvement; annealing algorithm; stochastic optimization; NP-hard combinatorial problems; travelling salesman; Steiner trees; convergence rate
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\textit{M. Lundy} and \textit{A. Mees}, Math. Program. 34, 111--124 (1986; Zbl 0581.90061)

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