Monday, January 19, 2009

Modern Genetic (And Other) Algorithms Explained: Part 4 - Simulated Annealing

(This is part 4 in the Modern Genetic Algorithms Explained series, click here to go to the first post, or browse through all the parts with the Table Of Contents at the end of this post.)

First of all: Simulated Annealing is not a genetic algorithm, but it is a modern optimization technique.

Wikipedia tells us the following:
Simulated annealing (SA) is a generic probabilistic meta-algorithm for the global optimization problem, namely locating a good approximation to the global minimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more effective than exhaustive enumeration — provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution.
The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become unstuck from their initial positions (a local minimum of the internal energy) and wander randomly through states of higher energy; the slow cooling gives them more chances of finding configurations with lower internal energy than the initial one.
Let's take a look at some pseudocode:

Randomly construct a valid solution
For each temperature do:
  For number of trials to do at each temperature do:
    Move solution to a neighbour
    Accept neighbour with probability(old_score, new_score, temperature)
    Lower the temperature by a reduction factor

It becomes clear that this method can be used in discrete optimization problems only, so that we can construct neighbours from our current state. E.g., a first solution in the problem we've been discussing might be [10,51,5,18] with a neighbour [10,52,5,18].

Also, a probability function needs to be defined. Some functions are constructed so that better solutions will always be accepted. Ideally, we would always like to assign a certain probability, so that worse solutions have a chance of being accepted too (or better ones have a chance at rejection). Again: this is to avoid local maxima (comparable with GA's).

Often, the new solution is accepted with an exponential distribution: exp( (new_score-old_score)/temperature ).

Note: the pseudocode at the Wikipedia page doesn't use trials, for some problems, this is good enough.

Let's see some Python code (again, based on the code mentioned in the previous posts):

from random import randint, random
from operator import add
from math import *

def individual(length, min, max):
  'Create a member of the population.'
  return [ randint(min,max) for x in xrange(length) ]

def fitness(individual, target):
  Determine the fitness of an individual. Higher is better.
  individual: the individual to evaluate
  target: the target number individuals are aiming for
  sum = reduce(add, individual, 0)
  return abs(target-sum)

def probability(o_fitness, n_fitness, temperature):
  if n_fitness < o_fitness:
    return 1
  return exp( (n_fitness-o_fitness) / temperature)

def temperature(step, max_steps):
  return max_steps - step

def neighbour(ind, min, max):
  pos_to_mutate = randint(0, len(ind)-1)

  if random() < 0.5:
    ind[pos_to_mutate] -= 1
    ind[pos_to_mutate] += 1

  if ind[pos_to_mutate] < min:
    ind[pos_to_mutate] = min
  elif ind[pos_to_mutate] > max:
    ind[pos_to_mutate] = max

  return ind

def evolve(ind, nr_trials, step, max_steps, min, max, target):
  best_fit = 10000;
  for i in range(1,nr_trials):
    n_ind = neighbour(ind, min, max)
    o_fitness = fitness(ind,target)
    n_fitness = fitness(n_ind,target)

  if n_fitness < best_fit:
    best_fit = n_fitness

  #move to new state?
  if probability(o_fitness, n_fitness, temperature(step,max_steps)) >= random():
    ind = n_ind
    print "Best fitness this evolution:",best_fit
    print "Temperature this evolution:",temperature(step,max_steps)

  return ind

If the fitness of the neighbour is better (remember: that means lower), we immediately accept it. We don't really need a chance of rejection for this problem. Otherwise, we use exp( (n_fitness-o_fitness) / temperature).
We use a separate function to calculate the temperature for each step. This function is kept fairly simple (max steps - this step), but non-linear temperature can also be implemented.

Let's try it, our starting temperature becomes 100, using 1000 trials per temperature:

import sys
from annealing import *
target = 300
i_length = 6
i_min = 0
i_max = 100
i = individual(i_length, i_min, i_max)
print fitness(i, target)
i_k = 0
i_kmax = 100
i_trials = 1000
while i_k < i_kmax:
  i = evolve(i, i_trials, i_k, i_kmax, i_min, i_max, target)
  i_k += 1

The output:
Best fitness this evolution: 34
Temperature this evolution: 100
Best fitness this evolution: 7
Temperature this evolution: 99
Best fitness this evolution: 0
Temperature this evolution: 98
Best fitness this evolution: 44
Temperature this evolution: 97
Best fitness this evolution: 43
Temperature this evolution: 96
Best fitness this evolution: 33
Temperature this evolution: 95
Best fitness this evolution: 39
Temperature this evolution: 94
Best fitness this evolution: 44
Temperature this evolution: 93
Best fitness this evolution: 34
Temperature this evolution: 92
Best fitness this evolution: 36
Temperature this evolution: 91
Best fitness this evolution: 50
Temperature this evolution: 90
Best fitness this evolution: 68
Temperature this evolution: 89
Best fitness this evolution: 67
Temperature this evolution: 88
Best fitness this evolution: 53
Temperature this evolution: 87
Best fitness this evolution: 49
Temperature this evolution: 86
Best fitness this evolution: 35
Temperature this evolution: 85
Best fitness this evolution: 55
Temperature this evolution: 84
Best fitness this evolution: 5
Temperature this evolution: 83
Best fitness this evolution: 0

Simulated annealing is easy to program and implement. Provided it is easy to construct neighbours, and a sensible combination of temperature and probability functions can be constructed, and the number of trials is well defined. Still: this method might be too naive to solve more difficult problems.

The source code can be downloaded here.

An interesting implementation is this. It is based on another genetic programming experiment located here (both are very interesting and fun examples, I highly recommend reading them).

Another Java applet to look at: solving a travelling salesman problem with simulated annealing.

Table Of Contents (click a link to jump to that post)

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