simulation::annealing(n)				       Tcl Simulation Tools					  simulation::annealing(n)

__________________________________________________________________________________________________________________________________________________

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:

       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 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
		     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  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:

       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 precedence 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  tem-
	      perature	-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:

       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  accept-
       ing 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 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 reduc-
       tion 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:

       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 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

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

simulation								0.2						  simulation::annealing(n)