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张英英

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Abstract: By combining the aspect of population in genetic algorithms (GAs) and the simulated annealing
algorithm (SAA),a novel algorithm, called fast annealing evolutionary algorithm (FAEA), is proposed. The
algorithm is similar to the annealing evolutionary algorithm (AEA), anda very fast annealing technique is
adopted for the annealing procedure. By an application of the algorithm to the optimization of test functions
anda comparison of the algorithm with other stochastic optimization methods, it is shown that the algorithm
isa highly efﬁcient optimization method. It was also applied in optimization of Lennard–Jones clusters and
compared with other methods in this study. The results indicate that the algorithm isa good tool for the
energy minimization problem.
© 2002 John Wiley& Sons, Inc. J Comput Chem 23: 427–435, 2002; DOI 10.1002/jcc.10029
Key words: annealing evolutionary algorithm; global optimization; Lennard–Jones clusters
Introduction
Determination of global energetic minima of large molecules or
atomic clusters is one of the challenging problems in computa-
tional chemistry due to the large number of local minima. Because
the number of local minima tends to grow exponentially with the
number of molecules or atoms, the task of minimizing the energy
of complicated molecules or atomic clusters is notoriously difﬁ-
cult. To solve the problem, different optimization methods have
been proposed, such as genetic algorithms,
1–5
simulated anneal-
ing,
6–8
quantum annealing,
9
potential deformation,
10
hierarchical
searches,
11
etc.
Genetic algorithms (GAs) are known asa new kind of optimi-
zation technique for tackling complicated optimization tasks.
12,13
A population of random bit (or digital) strings is used for starting
solution trials. Then,a circular process of evaluation, selection,
recombination, and mutation is repeated to yield an optimized
solution based upon the simulation of natural selection, genetics,
and evolution with Darwin’s principles of struggle for life and
survival of the ﬁttest. Therefore, GAs are based upon stochastic
search heuristics wherein an explorative and an exploitative mech-
anism cooperate, which is suitable to be used in optimizations of
complex hypersurface witha lot of local minima or maxima.
Within the past decade, applications of GAs to alloy systems,
14
atomic and molecular clusters,
15,16
molecular modeling,
17
protein
design,
18
molecular recognition,
19
protein folding,
20
and confor-
mation analysis
21
have been reported.
Simulated annealing algorithm (SAA) isa stochastic optimiza-
tion method based on the Monte Carlo importance-sampling tech-
nique.
22
It starts from an initial point and takesa single point
iterative strategy. The mechanism that it accept not only the
evolved but also the degenerated solutions with the Metropolis
acceptance criterion during its annealing procedure, makes the
SAA be of the potentiality to ﬁnd the global minimum instead of
falling into local minima. Asa global optimization algorithm, SAA
has been widely used to ﬁtting nonconvex cost functions arising in
a variety of problems, such as the ﬁtting of curves,
23
conformation
analysis,
24
analysis of atom–atom interaction,
6
and optimization of
molecular clusters.
8
In this articlea novel algorithm combining the aspect of pop-
ulation in GAs and simulated annealing procedure is proposed.
Because this algorithm is based on the annealing evolutionary
algorithm (AEA),
25
and the very fast annealing technique26,27
is
adopted for the annealing procedure, it is called the fast annealing
evolutionary algorithm (FAEA).
To assess the algorithm, we applied FAEA toa set of standard
test functions and compared the results with some other stochastic
methods. This is the usual procedure for all proposed optimization
methods. Furthermore, an application of the algorithm to the
optimization of Lennard–Jones (LJ) clusters
28–30
is also investi-
gated. It is shown that the algorithm isa good tool for energy
Correspondence to: W. Cai; e-mail: wscai@ustc.edu.cn
Contract/grant sponsor: Natural Science Foundation of China; contract/
grant number: 29975027
Contract/grant sponsor: Foundation of University of Science and
Technology of China for Young Scientists.
© 2002 John Wiley& Sons, Inc.minimization problems. Both the structure and the energy are in
good agreement with theoretical and calculated results in the
literature.
Method
Very Fast Annealing (VFA)
Simulated annealing was essentially introduced as a Monte Carlo
importance-sampling technique for doing large-dimension path
integrals arising in statistical physics problem. The method con-
sists of three functions, i.e., (1) g( x): probability density of state-
space of D parameters x { xi
; i 1, D}; (2) h( x): probability
density for acceptance of new cost-function given the just previous
value; (3) T(k): schedule of annealing the temperature T in an-
nealing-time steps k, i.e., of changing the volatility or ﬂuctuation
of the two previous probability densities.
Generally, h( x) 1/(1 exp(E/T )) exp(E/T), but
different g( x) and T(k) are used for different algorithms.
To statistically assure that any point in x-space can be sampled
inﬁnitely often in annealing time (IOT), it sufﬁces to prove that the
products of probabilities of not generating a state x IOT for all
annealing times successive to k0 yield zero,
26,27
k0

1 gk 0 (1)
This is equivalent to
k0

gk (2)
VFA was proposed by Ingber,
26,27
in which different parameter
has different annealing schedule. For a x-space of D parameters, a
new solution xi
[Ai
, Bi
] is generated by
xk1
i
xk
i
yi
Bi
Ai
(3)
where k is annealing-time index, and
yi
sgnui
1
2Ti1 1
Ti

2ui
1
1 (4)
where ui
U[0, 1] is a random number with uniform distribution.
The generating function corresponding to eq. (4) is
g y i1
D
1
2yi
Ti
ln1 1/Ti

(5)
If the annealing schedule of temperature T(k) is calculated by
Ti
k T0i
expci
k1/D (6)
the following equation can be obtained,
k0

gk k0

i1
D
1
2yi
ci

1
k
(7)
where ci
mi
exp(ni
/D), and mi
, ni
are parameters to control
the annealing schedule, which can be different for each speciﬁc
problem. To simplify the calculation, in this study, a same value of
mi
and ni
for every parameters is adopted.
To compare Boltzmann annealing (BA), fast annealing (FA)
and VFA, Boltzmann distribution used in BA, Cauchy distribution
used in FA and the generating function g( y) in eq. (5) were
illustrated in Figure 1, respectively. It is obvious that VFA has a
fattest tail among them, permitting easier access to test local
minima in the search for the desired global minimum. Due to the
character of the generating function g( y) and annealing schedule
T(k), the VFA will converge very fast.
26,27
But for complicated
optimization problem it inevitably converges to local minima.
Flowchart of Fast Annealing Evolutionary Algorithm
(FAEA)
Generally, an AEA combines the aspect of population and selec-
tion procedure in GAs and the annealing procedure in SAA. The
AEA improves the population of candidate solutions by selection
operation instead of the single-point iterative strategy used by the
SAA. Therefore, it has a larger possibility to escape from local
minima than the SAA.
25
But, in our experience, for complicated
optimization problem, such as the optimization of molecules or
atomic clusters, the AEA often converges at a local minimum. By
examination of the AEA calculations, it was found that the reason
for the early convergence may be caused by the selection opera-
tion. Although selection improves the ﬁtness of the population, it
will lose the diversity of the population. Therefore, in this study,
selection operation is not used in the FAEA algorithm. Further-
more, to make the algorithm converge effectively and guarantee
that the algorithm ﬁnds the global minimum, VFA technique,
similarity checks between individuals in population, and a local
search procedure are adopted. The whole procedure of the FAEA
is shown in Figure 2, and the following steps are included.
1. Prepare parameters for the algorithm, which include the m, n,
ns (length of Markov chain), and initial temperature (T0) for
the annealing schedule, population size (npop) for evolution,
the number of parameters or atoms in clusters (N), and the
range of each parameter. The stop criterion of the algorithm,
Texit
, will also be calculated by T0exp(m).
26,27
2. According to the atom number and population size, the pop-
ulation is randomly initialized and evaluated. The value of
every gene in each individual is generated by u(B A),
where u U[0, 1] is a random number with uniform dis-
tribution, and A, B are the range of the optimizing parameters.
3. According to eq. (6), calculate the current temperature T.
4. As discussed above, VFA converges very fast. It is important
to keep diversity in the population in order to avoid converg-
ing to local minima. Therefore, in the FAEA, a procedure to
check similarity between individuals is developed, i.e., during
428 Cai and Shao • Vol. 23, No. 4 • Journal of Computational Chemistry

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