Page 30 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

18 Chapter 1

After Crossover

Generate the random number r=0.3(say)

C1=r*C1+(1-r)*C2= [0.2680 0.3590 0.2430]
C2=(1-r)*C1+r*C2= [0.2920 0.4110 0.1670 ]

3. SIMULATED ANNEALING

The process of heating the solid body to the high temperature and allowed to
cool slowly is called Annealing. Annealing makes the particles of the solid
material to reach the minimum energy state. This is due to the fact that when
the solid body is heated to the very high temperature, the particles of the
solid body are allowed to move freely and when it is cooled slowly, the
particles are able to arrange itself so that the energy of the particles are made
minimum. The mathematical equivalent of the thermodynamic annealing as
described above is called simulated annealing.
The energy of the particle in thermodynamic annealing process can be
compared with the cost function to be minimized in optimization problem.
The particles of the solid can be compared with the independent variables
used in the minimization function.
Initially the values assigned to the variables are randomly selected from
the wide range of values. The cost function corresponding to the selected
values are treated as the energy of the current state. Searching the values
from the wide range of the values can be compared with the particles
flowing in the solid body when it is kept in high temperature.
The next energy state of the particles is obtained when the solid body is
slowly cooled. This is equivalent to randomly selecting next set of the
values.
When the solid body is slowly cooled, the particles of the body try to
reach the lower energy state. But as the temperature is high, random flow of
the particles still continuous and hence there may be chance for the particles
to reach higher energy state during this transition. Probability of reaching the
higher energy state is inversely proportional to the temperature of the solid
body at that instant.
In the same fashion the values are randomly selected so that cost function
of the currently selected random values is minimum compared with the
previous cost function value. At the same time, the values corresponding to
the higher cost function compared with the previous cost function are also
selected with some probability. The probability depends upon the current
simulated temperature ‘T’. If the temperature is large, probability of