simulation::annealing(n) Tcl Simulation Tools simulation::annealing(n)______________________________________________________________________________NAME
simulation::annealing - Simulated annealingSYNOPSIS
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 neces‐
sarily 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 com‐
mand.
string keyword
Given keyword (without leading minus)
::simulation::annealing::hasOption keyword
Returns 1 if the option is available, 0 if not.
string keyword
Given keyword (without leading minus)
::simulation::annealing::setOption keyword value
Set the value of the given option.
string keyword
Given keyword (without leading minus)
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 key‐
word-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 simu‐
lated 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 prece‐
dence 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 fac‐
tor (between 0 and 1) -initial-temp t: initial temperature
-scale s: scale of the function (order of magnitude of the val‐
ues) -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 fac‐
tor 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 func‐
tion to be minimized depends on all or most parameters - see the
source code for a counter example and run that.
KEYWORDS
math, optimization, simulated annealingCATEGORY
Mathematics
COPYRIGHT
Copyright (c) 2008 Arjen Markus <arjenmarkus@users.sourceforge.net>
simulation 0.2 simulation::annealing(n)
