#!/usr/bin/env python
# Python module for simulated annealing - anneal.py - v1.0 - 2 Sep 2009
#
# Copyright (c) 2009, Richard J. Wagner <wagnerr@umich.edu>
#
# Permission to use, copy, modify, and/or distribute this software for any
# purpose with or without fee is hereby granted, provided that the above
# copyright notice and this permission notice appear in all copies.
#
# THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
# WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
# MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
# ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
# WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
# ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
# OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
import logging as log
"""
This module performs simulated annealing to find a state of a system that
minimizes its energy.
An example program demonstrates simulated annealing with a traveling
salesman problem to find the shortest route to visit the twenty largest
cities in the United States.
"""
# How to optimize a system with simulated annealing:
#
# 1) Define a format for describing the state of the system.
#
# 2) Define a function to calculate the energy of a state.
#
# 3) Define a function to make a random change to a state.
#
# 4) Choose a maximum temperature, minimum temperature, and number of steps.
#
# 5) Set the annealer to work with your state and functions.
#
# 6) Study the variation in energy with temperature and duration to find a
# productive annealing schedule.
#
# Or,
#
# 4) Run the automatic annealer which will attempt to choose reasonable values
# for maximum and minimum temperatures and then anneal for the allotted time.
import copy
import math
import random
import sys
import time
import numpy as np
[docs]def time_string(seconds):
"""Returns time in seconds as a string formatted HHHH:MM:SS."""
s = int(round(seconds)) # round to nearest second
h, s = divmod(s, 3600) # get hours and remainder
m, s = divmod(s, 60) # split remainder into minutes and seconds
return "%4i:%02i:%02i" % (h, m, s)
[docs]class Annealer:
"""Performs simulated annealing by calling functions to calculate
energy and make moves on a state. The temperature schedule for
annealing may be provided manually or estimated automatically.
"""
def __init__(self, energy, move):
self.energy = energy # function to calculate energy of a state
self.move = move # function to make a random change to a state
[docs] def anneal(self, state, Tmax, Tmin, steps, updates=0):
"""Minimizes the energy of a system by simulated annealing.
Keyword arguments:
state -- an initial arrangement of the system
Tmax -- maximum temperature (in units of energy)
Tmin -- minimum temperature (must be greater than zero)
steps -- the number of steps requested
updates -- the number of updates to print during annealing
Returns the best state and energy found."""
step = 0
start = time.time()
def update(T, E, acceptance, improvement):
"""Prints the current temperature, energy, acceptance rate,
improvement rate, elapsed time, and remaining time.
The acceptance rate indicates the percentage of moves since the last
update that were accepted by the Metropolis algorithm. It includes
moves that decreased the energy, moves that left the energy
unchanged, and moves that increased the energy yet were reached by
thermal excitation.
The improvement rate indicates the percentage of moves since the
last update that strictly decreased the energy. At high
temperatures it will include both moves that improved the overall
state and moves that simply undid previously accepted moves that
increased the energy by thermal excititation. At low temperatures
it will tend toward zero as the moves that can decrease the energy
are exhausted and moves that would increase the energy are no longer
thermally accessible."""
elapsed = time.time() - start
if step == 0:
log.info(
" Temperature Energy Accept Improve Elapsed Remaining Step"
)
log.info(
"%12.2e %12.2e %s "
% (T, E, time_string(elapsed))
)
else:
remain = (steps - step) * (elapsed / step)
log.info(
"%12.2e %12.2e %7.2f%% %7.2f%% %s %s %s"
% (
T,
E,
100.0 * acceptance,
100.0 * improvement,
time_string(elapsed),
time_string(remain),
step,
)
)
# Precompute factor for exponential cooling from Tmax to Tmin
if Tmin <= 0.0:
log.info(
"Exponential cooling requires a minimum temperature greater than zero."
)
sys.exit()
Tfactor = -math.log(float(Tmax) / Tmin)
# Note initial state
T = Tmax
E = self.energy(state)
prevState = state.copy()
prevEnergy = E
bestState = state.copy()
bestEnergy = E
trials, accepts, improves = 0, 0, 0
if updates > 0:
updateWavelength = updates
update(T, E, None, None)
# Attempt moves to new states
while step < steps:
step += 1
T = Tmax * math.exp(Tfactor * step / steps)
self.move(state)
E = self.energy(state)
dE = E - prevEnergy
trials += 1
if dE > 0.0 and math.exp(-dE / T) < random.random():
# Restore previous state
state = prevState.copy()
E = prevEnergy
else:
# Accept new state and compare to best state
accepts += 1
if dE < 0.0:
improves += 1
prevState = state.copy()
prevEnergy = E
if E < bestEnergy:
bestState = state.copy()
bestEnergy = E
if updates > 1:
if step // updateWavelength > (step - 1) // updateWavelength:
update(T, E, float(accepts) / trials, float(improves) / trials)
trials, accepts, improves = 0, 0, 0
# Return best state and energy
return bestState, bestEnergy
[docs] def auto(self, state, minutes, steps=2000):
"""Minimizes the energy of a system by simulated annealing with
automatic selection of the temperature schedule.
Keyword arguments:
state -- an initial arrangement of the system
minutes -- time to spend annealing (after exploring temperatures)
steps -- number of steps to spend on each stage of exploration
Returns the best state and energy found."""
def run(state, T, steps):
"""Anneals a system at constant temperature and returns the state,
energy, rate of acceptance, and rate of improvement."""
E = self.energy(state)
prevState = state.copy()
prevEnergy = E
accepts, improves = 0, 0
for step in range(steps):
self.move(state)
E = self.energy(state)
dE = E - prevEnergy
if dE > 0.0 and math.exp(-dE / T) < random.random():
state = prevState.copy()
E = prevEnergy
else:
accepts += 1
if dE < 0.0:
improves += 1
prevState = state.copy()
prevEnergy = E
return state, E, float(accepts) / steps, float(improves) / steps
step = 0
start = time.time()
log.info("Attempting automatic simulated anneal...")
# Find an initial guess for temperature
T = 0.0
E = self.energy(state)
while T == 0.0:
step += 1
self.move(state)
T = abs(self.energy(state) - E)
log.info("Exploring temperature landscape:")
log.info(" Temperature Energy Accept Improve Elapsed")
def update(T, E, acceptance, improvement):
"""Prints the current temperature, energy, acceptance rate,
improvement rate, and elapsed time."""
elapsed = time.time() - start
log.info(
"%12.2e %12.2e %7.2f%% %7.2f%% %s"
% (T, E, 100.0 * acceptance, 100.0 * improvement, time_string(elapsed))
)
# Search for Tmax - a temperature that gives 98% acceptance
state, E, acceptance, improvement = run(state, T, steps)
step += steps
while acceptance > 0.98:
T = round_figures(T / 1.5, 2)
state, E, acceptance, improvement = run(state, T, steps)
step += steps
update(T, E, acceptance, improvement)
while acceptance < 0.98:
T = round_figures(T * 1.5, 2)
state, E, acceptance, improvement = run(state, T, steps)
step += steps
update(T, E, acceptance, improvement)
Tmax = T
# Search for Tmin - a temperature that gives 0% improvement
while improvement > 0.0:
T = round_figures(T / 1.5, 2)
state, E, acceptance, improvement = run(state, T, steps)
step += steps
update(T, E, acceptance, improvement)
Tmin = T
# Calculate anneal duration
elapsed = time.time() - start
duration = round_figures(int(60.0 * minutes * step / elapsed), 2)
# Perform anneal
log.info("Annealing from %.2f to %.2f over %i steps:" % (Tmax, Tmin, duration))
return self.anneal(state, Tmax, Tmin, duration, 20)