#tex2html_wrap_inline1774#) of
the SA algorithm using:
#equation1525#
Typically the value of #tex2html_wrap_inline1776# is chosen in the range 0.90 to 0.99.
This cooling schedule has the advantage of being well understood, having
a solid theoretical foundation, and being the most widely used annealing
schedule.
Our results obtained from using this standard cooling schedule will be used
as a baseline for comparison with those using the second and third schedules.
The second annealing schedule we used is the method of reheating as a
function of cost (RFC) [#2##1#]. Before introducing this schedule we
first summarize a few relevant points on the concept of specific heat
(#tex2html_wrap_inline1782#). Specific heat is a measure of the variance of the cost (or
energy) values of states at a given temperature. The higher the
variance, the longer it presumably takes to reach equilibrium, and so
the longer one should spend at the temperature, or alternatively, the
slower one should lower the temperature.
Generally, in combinatorial optimization problems, phase transitions
[#8##1#, #66##1#] can be observed as sub-parts of the problem are resolved.
In some of the work dealing with the TSP using annealing,
e.g. [#7##1#], the authors often observe that the resolution of the
overall structure of the solution occurs at high temperatures, and at
low temperatures the fine details of the solution are resolved.
As reported in [#2##1#], applying a reheating type procedure,
depending on the phase, would allow the algorithm to spend more time
in the low temperature phases, thus reducing the total amount of time
required to solve a given problem.
In order to calculate the temperature at which a phase transition
occurs, it is necessary to compute the specific heat of the
system. A phase transition occurs at a temperature #tex2html_wrap_inline1784# when
the specific heat is maximal (#tex2html_wrap_inline1786#), and this triggers the change in
the state ordering. If the best solution found to date has a high
energy or cost then the super-structure may require
re-arrangement. This can be done by raising the temperature to a level
which is higher than the phase transition temperature #tex2html_wrap_inline1788#.
Generally, the higher the current best cost, the higher the
temperature which is required to escape the local minimum.
Now, to compute the aforementioned maximum specific heat we employ the
following steps which we refer to as the RFC schedule. The steps
involved in this schedule are mostly adapted from the work of van
Laarhoven and Aarts [#17##1#] and Abramson et al. [#2##1#].
At each temperature T, the annealing algorithm generates a set of
configurations #tex2html_wrap_inline1792#.
Let #tex2html_wrap_inline1794# denote the cost of configuration i, C(T) is the average
cost at temperature T, and #tex2html_wrap_inline1802# is the standard deviation of the
cost at T.
At temperature T, the acceptance probability is:
#equation1528#
The average cost is computed as:
#equation1531#
Therefore, the average square cost is:
#equation1534#
The variance of the cost is:
#equation1537#
Now, the specific heat is defined as:
#equation1540#
The temperature #tex2html_wrap_inline1808# at which the maximum specific heat
occurs, or at which the system undergoes a phase transition, can thus
be found.
Reheating sets the new temperature to be:
#equation1543#
where K is a constant and #tex2html_wrap_inline1812# is the current best cost.
It is worth pointing out that reheating is done when the temperature
drops below the phase transition (the point of maximum specific heat)
and there has been no decrease in cost for a specified number of
iterations, i.e. the system gets stuck in a local minimum. Reheating
increases the temperature above the phase transition (see equation
#T6#464>), in order to produce enough of a change in the configuration
to allow it to explore other minima when the temperature is reduced again.
The third cooling (adaptive) schedule used is to compute a
new temperature based not on the best cost found so far but the
standard deviation of all costs obtained at the current T. A similar
approach was taken by Huang et al. [#59##1#] yielding an efficient
cooling schedule. Let #tex2html_wrap_inline1816# denote the current
temperature, at step j of the annealing schedule.
After calculating #tex2html_wrap_inline1820# from equation #T4#466>, the new T
which is denoted as #tex2html_wrap_inline1824# is as follows:
#equation1546#
where a is a parameter that we have set to 0.0005.
Following suggestions by Otten and van Ginneken [#60##1#] and Diekmann
et al. [#61##1#], #tex2html_wrap_inline1830# is smoothed out in order to avoid any
dependencies of the temperature decrement on large changes in the
standard deviation #tex2html_wrap_inline1832#.
We used the following standard method to provide a smoothed standard
deviation #tex2html_wrap_inline1834#:
#equation1549#
and set #tex2html_wrap_inline1836# to 0.95.
This smoothing function is used because it follows (from the form of the
Boltzmann distribution, see [#41##1#, #13##1#]) that it preserves the key
relationship:
#equation1552#
We would like to point out that the ``smoothing'' approach of
equation #T60#482> is similar to an approach used in iterative
methods for sparse linear systems. Specifically, the parameter
#tex2html_wrap_inline1840# is used to compute the successive over-relaxation (or SOR)
phase steps in the Gauss-Seidel algorithm where the former is an
extension of the later method [#65##1#].