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Comparison of Di erent Neighbourhood Sizes in Simulated Annealing Xin Yao Department of Computer Science University College, University of New South Wales Australian Defence Force Academy Canberra, ACT, Australia 2600

Abstract
Neighbourhood structure and size are important parameters in local search algorithms. This is also true for generalised local search algorithms like simulated annealing. It has been shown that the performance of simulated annealing can be improved by adopting a suitable neighbourhood size. However, previous studies usually assumed that the neighbourhood size was xed during search. This paper presents a simulated annealing algorithm with a dynamic neighbourhood size which depends on the current \temperature" value during search. A method of dynamically deciding the neighbourhood size by approximating a continuous probability distribution is given. Four continuous probability distributions are used in our experiments to generate neighbourhood sizes dynamically, and the results are compared.

combinatorial optimisation. A method of generating dynamic neighbourhood sizes by approximating continuous probability distributions is given in this section. Section 4 compares the experimental results of using di erent continuous probability distributions to generate dynamic neighbourhood sizes. Finally, Section 5 concludes with some remarks and directions of future research.

2 General Simulated Annealing
Although SA can be used in both continuous and discrete cases, this paper only considers combinatorial optimisation by SA unless otherwise indicated explicitly. A combinatorial optimisation problem can be informally described as nding an optimal con guration X from a nite or in nite countable con guration space S . Each con guration X 2 S can be represented by its n (> 0) components, i.e., X = (x1; x2; ; xn ), where xi 2 Xi , i = 1; 2; ; n. An excellent discussion of combinatorial optimisation and its complexity can be found in Garey and Johnson's book 8]. A general model of SA, which is applicable to both continuous and discrete problems, can be described by Figure 1, where function generate (X; Tn) is decided by the generation probability gXY (Tn ), which is the probability of generating con guration Y from con guration X at temperature Tn , function accept (X; Y; Tn) is decided by the acceptance probability aXY (Tn ), which is the probability of accepting con guration Y after it has been generated at temperature Tn , and function update (Tn ) decides the rate of the temperature decrease. These three functions determine the convergence of general SA 5, 6, 9], but parameters in general SA, such as the initial temperature, initial con guration, inner-loop stop criterion, and outer1

1 Introduction
Simulated Annealing (SA) algorithms can nd very good near optimal solutions to a wide range of hard problems, but at the high computational cost. Various methods have been proposed to speed up its convergence, which can roughly be divided into three categories: (1) Optimising functions and parameters in SA 1]; (2) Combining SA with other search algorithms 2, 3]; and (3) Parallelising SA 4]. This paper falls into the above rst category. Section 2 of this paper describes a general SA algorithm 5, 6] which uni es di erent variants of the classical one 7]. Section 3 presents SA with a dynamic neighbourhood size and its application in Published in Proc. of Fourth Australian Conf. on Neural Networks, ed. P. Leong and M. Jabri, pp.216{219, 1993, Melbourne, Australia.

generate initial con guration X at random; generate initial temperature T0; REPEAT REPEAT Y = generate(X; Tn); IF accept(X; Y; Tn) THEN X = Y ; UNTIL `inner-loop stop criterion' satis ed; Tn+1 = update (Tn ); n = n + 1; UNTIL `outer-loop stop criterion' satis ed Figure 1: General simulated annealing. loop stop criterion, can have signi cant impact on its nite-time behaviour. That is, the computation time in practice depends on the three functions...
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