NAME

AI::SimulatedAnnealing - optimize a list of numbers according to a specified cost function.

SYNOPSIS

use AI::SimulatedAnnealing;

$optimized_list = anneal(
  $number_specs, $cost_function, $cycles_per_temperature);

DESCRIPTION

This module provides a single public function, anneal(), that optimizes a list of numbers according to a specified cost function.

Each number to be optimized has a lower bound, an upper bound, and a precision, where the precision is an integer in the range 0 to 4 that specifies the number of decimal places to which all instances of the number will be rounded. The upper bound must be greater than the lower bound but not greater than 10 to the power of (4 - p), where "p" is the precision. The lower bound must be not less than -1 times the result of taking 10 to the power of (4 - p).

A bound that has a higher degree of precision than that specified for the number to which the bound applies is rounded inward (that is, downward for an upper bound and upward for a lower bound) to the nearest instance of the specified precision.

The attributes of a number (bounds and precision) are encapsulated within a number specification, which is a reference to a hash containing "LowerBound", "UpperBound", and "Precision" fields.

The anneal() function takes a reference to an array of number specifications, a cost function, and a positive integer specifying the number of randomization cycles per temperature to perform. The anneal() function returns a reference to an array having the same length as the array of number specifications. The returned list represents the optimal list of numbers matching the specified attributes, where "optimal" means producing the lowest cost.

The cost function must take a reference to an array of numbers that match the number specifications. The function must return a single number representing a cost to be minimized.

In order to work efficiently with the varying precisions, the anneal() function converts each bound to an integer by multiplying it by 10 to the power of the precision; then the function performs the temperature reductions and randomization cycles (which include tests performed via calls to the cost function) on integers in the resulting ranges. When passing an integer to the cost function or when storing the integer in a collection of numbers to be returned by the function, anneal() first converts the integer back to the appropriate decimal number by dividing the integer by 10 to the power of the precision.

The initial temperature is the size of the largest range after the bounds have been converted to integers. During each temperature reduction, the anneal() function multiplies the temperature by 0.95 and then rounds the result down to the nearest integer (if the result isn't already an integer). When the temperature reaches zero, annealing is immediately terminated.

NOTE:  Annealing can sometimes complete before the temperature
reaches zero if, after a particular temperature reduction, a
brute-force optimization approach (that is, testing every possible
combination of numbers within the subranges determined by the new
temperature) would produce a number of tests that is less than or
equal to the specified cycles per temperature.  In that case, the
anneal() function performs the brute-force optimization to complete
the annealing process.

After a temperature reduction, the anneal() function determines each new subrange such that the current optimal integer from the total range is as close as possible to the center of the new subrange. When there is a tie between two possible positions for the subrange within the total range, a "coin flip" decides.

PREREQUISITES

This module requires Perl 5, version 5.10.1 or later.

METHODS

anneal($number_specs, $cost_function, $cycles_per_temperature);

The anneal() function takes a reference to an array of number specifications (which are references to hashes containing "LowerBound", "UpperBound", and "Precision" fields), a code reference pointing to a cost function (which takes a list of numbers matching the specifications and returns a number representing a cost to be minimized), and a positive integer specifying the number of randomization cycles to perform at each temperature.

The function returns a reference to an array containing the optimized list of numbers.

AUTHOR

Benjamin Fitch, <blernflerkl@yahoo.com>

COPYRIGHT AND LICENSE

Copyright 2010 by Benjamin Fitch.

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.