Procedure [22]
The simulated annealing method simulates the process of slow cooling of molten metal to achieve the minimum function value in a minimization prob- lem。 The cooling phenomenon of the molten metal is simulated by introducing a temperature-like parame- ter and controlling it using the concept of Boltzmann’s probability distribution。 The Boltzmann’s probability distribution implies that the energy (E) of a system in
thermal equilibrium at temperature T is distributed probabilistically according to the relation:
E
P (E ) e kT (46)
where P(E) denotes the probability of achieving the energy level E, and k is called the Boltzmann’s cons- tant。 Equation (46) shows that at high temperatures the system has nearly a uniform probability of being at any energy state。 However, at low temperatures, the system has a small probability of being at a high- energy state。 This indicates that when the search process is assumed to follow Boltzmann’s probability distribution, the convergence of the simulated anneal- ing algorithm can be controlled by controlling the tem- perature T。 The method of implementing the Boltz-
mann’s probability distribution in simulated thermody-
Note that the probability of accepting the point Xi+1 is not same in all situations。 As can be seen from Eq。 (50), this probability depends on the values of ∆E and T。 If the temperature T is large, the probability will be high for design points Xi+1 with larger function values (with larger values of ∆E = ∆f)。 Thus at high tempe- ratures, even worse design points Xi+1 are likely to be accepted because of larger probabilities。 However, if the temperature T is small, the probability of accept- ing worse design points Xi+1 (with larger values of ∆E =
= ∆f) will be small。 Thus as the temperature values
get smaller (that is, as the process gets closer to the optimum solution), the design points Xi+1 with larger function values compared to the one at Xi are less likely to be accepted。
ΔE
namic systems, suggested by Metropolis et al。 [18]
P (E i 1) e
kT (50)
can also be used in the context of minimization of functions。 In the case of function minimization, let the current design point (state) be Xi, with the corres- ponding value of the objective function given by fi =
= f(Xi)。 Similar to the energy state of a thermodynamic
system, the energy Ei at state Xi is given by:
Ei fi f (X i ) (47)
Then, according to the Metropolis criterion, the pro- bability of the next design point (state) Xi+1 depends on the difference in the energy state or function va- lues at the two design points (states) given by: 模拟退火技术来设计英文文献和中文翻译(9):http://www.youerw.com/fanyi/lunwen_100221.html