Randomization in Discrete Optimization: Annealing Algorithms", in Encyclopedia of Optimization, edited by P.M. Pardalos

Annealing algorithms have been employed extensively in the past decade to solve myriads of op-timization problems. Several intractable problems such as the traveling salesperson problem, graph partitioning, circuit layout, etc. have been solved to get satisfactory results. In this article we survey convergence results known for annealing algorithms. In particular we deal with Simulated Annealing and Nested Annealing. Simulated Annealing (SA) is a randomized heuristic that can be used to solve any combinatorial optimization problem. SA is typically used to produce quasi-optimal results. In practice SA has been applied to solve some presumably hard (e.g., NP-hard) problems. The level of performance obtained has been promising (Golden and Skiscim 1986, ElGamal and Shperling April 1984, Johnson et al. 1987, Vecchi and Kirkpatrick 1982). The success of this heuristic technique has motivated the study of convergence of this technique. One of the early results in this direction is due to Mitra et al. (Sept. 1986)) who proved that SA converges in the limit to a globally optimal solution with probability 1. Later results proved certain time bounds within which SA is guaranteed to converge with high probability. In this article we provide one such proof (due to Rajasekaran (2000)). Nested Annealing (NA) (Rajasekaran and Reif 1992) is a variation of SA that has been proven to perform better for optimization problems for which the cost function has some special properties (Rajasekaran and Reif 1992). In this article we provide a summary of NA and its convergence properties.