Provided by: tcllib_1.14-dfsg-1_all #### NAME

```       simulation::annealing - Simulated annealing

```

#### SYNOPSIS

```       package require Tcl  ?8.4?

package require simulation::annealing  0.2

::simulation::annealing::getOption keyword

::simulation::annealing::hasOption keyword

::simulation::annealing::setOption keyword value

::simulation::annealing::findMinimum args

::simulation::annealing::findCombinatorialMinimum args

_________________________________________________________________

```

#### DESCRIPTION

```       The  technique of simulated annealing provides methods to estimate the global optimum of a
function. It is described in some detail on the Wiki http://wiki.tcl.tk/.... The  idea  is
simple:

·      randomly select points within a given search space

·      evaluate the function to be optimised for each of these points and select the point
that has the lowest (or highest) function value or - sometimes  -  accept  a  point
that  has  a  less  optimal  value. The chance by which such a non-optimal point is
accepted diminishes over time.

·      Accepting less optimal points means the method does not necessarily get stuck in  a
local  optimum and theoretically it is capable of finding the global optimum within
the search space.

The method resembles the cooling of material, hence the name.

The package simulation::annealing offers the command findMinimum:

puts [::simulation::annealing::findMinimum  -trials 300  -parameters {x -5.0 5.0 y -5.0 5.0}  -function {\$x*\$x+\$y*\$y+sin(10.0*\$x)+4.0*cos(20.0*\$y)}]

prints    the    estimated    minimum    value    of     the     function     f(x,y)     =
x**2+y**2+sin(10*x)+4*cos(20*y) and the values of x and y where the minimum was attained:

result -4.9112922923 x -0.181647676593 y 0.155743646974

```

#### PROCEDURES

```       The package defines the following auxiliary procedures:

::simulation::annealing::getOption keyword
Get the value of an option given as part of the findMinimum command.

string keyword

::simulation::annealing::hasOption keyword
Returns 1 if the option is available, 0 if not.

string keyword

::simulation::annealing::setOption keyword value
Set the value of the given option.

string keyword

string value
(New) value for the option

The main procedures are findMinimum and findCombinatorialMinimum:

::simulation::annealing::findMinimum args
Find  the  minimum  of  a  function using simulated annealing. The function and the
method's parameters is given via a list of keyword-value pairs.

int n  List of keyword-value pairs, all of which are available during the execution
via the getOption command.

::simulation::annealing::findCombinatorialMinimum args
Find the minimum of a function of discrete variables using simulated annealing. The
function and the method's parameters is given via a list of keyword-value pairs.

int n  List of keyword-value pairs, all of which are available during the execution
via the getOption command.

The findMinimum command predefines the following options:

·      -parameters list: triples defining parameters and ranges

·      -function expr: expression defining the function

·      -code  body: body of code to define the function (takes precedence over -function).
The code should set the variable "result"

·      -init code: code to be run at start up -final code: code  to  be  run  at  the  end
-trials  n: number of trials before reducing the temperature -reduce factor: reduce
the temperature  by  this  factor  (between  0  and  1)  -initial-temp  t:  initial
temperature  -scale  s:  scale  of  the function (order of magnitude of the values)
-estimate-scale y/n: estimate the scale (only if -scale is  not  present)  -verbose
y/n:  print  detailed  information  on  progress  to the report file (1) or not (0)
-reportfile file: opened file to print to (defaults to stdout)

Any  other  options  can  be  used  via  the  getOption  procedure  in  the   body.    The
findCombinatorialMinimum command predefines the following options:

·      -number-params n: number of binary parameters (the solution space consists of lists
of 1s and 0s). This is a required option.

·      -initial-values: list of 1s and 0s constituting the start of the search.

The other predefined options are identical to those of findMinimum.

```

#### TIPS

```       The procedure findMinimum works by constructing a temporary procedure that does the actual
work.  It loops until the point representing the estimated optimum does not change anymore
within the given number of trials. As the temperature gets lower and lower the  chance  of
accepting a point with a higher value becomes lower too, so the procedure will in practice
terminate.

It is possible to optimise over a non-rectangular region, but some care must be taken:

·      If the point is outside the region of interest, you can specify a very high value.

·      This does mean that the automatic determination of a scale factor  is  out  of  the
question  -  the  high function values that force the point inside the region would
distort the estimation.

Here is an example of finding an optimum inside a circle:

puts [::simulation::annealing::findMinimum  -trials 3000  -reduce 0.98  -parameters {x -5.0 5.0 y -5.0 5.0}  -code {
if { hypot(\$x-5.0,\$y-5.0) < 4.0 } {
set result [expr {\$x*\$x+\$y*\$y+sin(10.0*\$x)+4.0*cos(20.0*\$y)}]
} else {
set result 1.0e100
}
}]

The method is theoretically capable of determining the global optimum, but often you  need
to  use  a  large number of trials and a slow reduction of temperature to get reliable and
repeatable estimates.

You can use the -final option to use a deterministic optimization  method,  once  you  are
sure you are near the required optimum.

The  findCombinatorialMinimum procedure is suited for situations where the parameters have
the values 0 or 1 (and there can be many of them). Here is an example:

·      We have a function that attains an absolute minimum if the first ten numbers are  1
and the rest is 0:

proc cost {params} {
set cost 0
foreach p [lrange \$params 0 9] {
if { \$p == 0 } {
incr cost
}
}
foreach p [lrange \$params 10 end] {
if { \$p == 1 } {
incr cost
}
}
return \$cost
}

·      We  want  to  find  the solution that gives this minimum for various lengths of the
solution vector params:

foreach n {100 1000 10000} {
break
puts "Problem size: \$n"
puts [::simulation::annealing::findCombinatorialMinimum  -trials 300  -verbose 0  -number-params \$n  -code {set result [cost \$params]}]
}

·      As the vector grows, the computation time increases, but the procedure will stop if
some  kind  of equilibrium is reached. To achieve a useful solution you may want to
try different values of the trials parameter for instance.  Also  ensure  that  the
function  to  be  minimized depends on all or most parameters - see the source code
for a counter example and run that.

```

#### KEYWORDS

```       math, optimization, simulated annealing

```

#### CATEGORY

```       Mathematics

```

```       Copyright (c) 2008 Arjen Markus <arjenmarkus@users.sourceforge.net>