


simulation::annealing(nTcl Simulation Toosimulation::annealing(n)


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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

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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:

       o      randomly select points within a given search space

       o      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.

       o      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:



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simulation::annealing(nTcl Simulation Toosimulation::annealing(n)


       ::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:

       o      -parameters list: triples defining parameters and ranges

       o      -function expr: expression defining the function

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

       o      -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



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simulation::annealing(nTcl Simulation Toosimulation::annealing(n)


              factor  (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:

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

       o      -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:

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

       o      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.




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simulation::annealing(nTcl Simulation Toosimulation::annealing(n)


       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:

       o      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
              }


       o      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]}]
              }


       o      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 annealing

CATEGORY
       Mathematics

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








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