![](bg3.png)
4 SIMULATED ANNEALING
ALGORITHM
Simulated annealing algorithm is a heuristic
algorithm designed to randomly search the global
optimal solution in the feasible solution space by
combining probabilistic jump characteristics.
If the new feasible solution
is found to be
better than the current feasible solution
, the new
feasible solution is accepted. Otherwise, the
Metropolis criterion determines whether to accept
the new feasible solution. In order not to reject
directly, define the acceptance probability
.
lies between [0,1], and measures the distance
between
and
. The closer is the
distance, the larger is
. Here we make
assumptions.
exp
ji
P f x f x
(20)
In order to improve the efficiency of the
algorithm, in the early stage of the algorithm search,
it is necessary to improve the scope of the algorithm
search to avoid falling into local optimal. In the later
stage of the search, it is necessary to reduce the
search scope of the algorithm as much as possible.
That is, it just searches locally, because at this time
it is close to the global optimal. We make a
deformation of the above formula (20).
exp
t j i
P C f x f x
(21)
in the formula (21) can be regarded as a time-
dependent coefficient. Then the probability P of the
algorithm accepting the new feasible solution
establishes a relationship with the time parameter.
If t is small in the early stage of search, and the
search scope is large enough, then the corresponding
P needs to be larger. And
is set to be negatively
correlated with
, so it should be small. If
is
smaller in the late search period,
should be
larger. Obviously, the longer time goes, the bigger
gets.
The flow of the search process is as follows.
1) Generate an initial solution A randomly, and
calculate the objective function
corresponding to the initial solution.
2) A solution B is generated near the initial
solution according to the probability mechanism,
and the objective function
corresponding to
the new solution B is calculated.
3) If
, the new solution overwrites
the original solution and repeat the above steps.
If
, it calculates the probability of
accepting the newer solution B, that is
exp .
tt
P f B f A C
Then it randomly
generates number
. If
, the initial
solution A is overwritten by the new solution B. And
the above steps are repeated. Otherwise, it returns to
the second step. A newer solution
is re-generate
near the initial solution, and it continues to iterate.
However, there is a problem in the above
process, that is, the setting of key coefficient
. So
we define the initial temperature
.
According to thermodynamics, the formula for
temperature drop is
(22)
In the formula (22),
is usually 0.95, then the
temperature at time t is
(23)
To ensure that
increases about t, we have
(24)
Then
exp exp
100 0.95
t
t
t
f B f A f B f A
P
T
(25)
Let
, when the temperature
is constant, the smaller
is, the greater the
probability
is. That is, the smaller the difference
from the existing solution is, the greater the
possibility of accepting the newer solution is. When
is constant, the higher the temperature is, the
greater the acceptance probability is. Therefore, it is
easier to accept the newer solution when the
temperature is high in the early stage of search.
5 SIMULATION CALCULATION
The theoretical basis of Monte Carlo method is the
law of large numbers. The law of large number
describes the results of a considerable number of
repeated experiments, and according to this law, the
larger the number of samples, the closer the average
will be to the true value.
ANIT 2023 - The International Seminar on Artificial Intelligence, Networking and Information Technology
554