anneal.py

#!/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.

"""
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.

Notes:
    Matt Perry 6/24/12 
        Changed to slicing lists instead of deepcopy-ing them.
        e.g. state = prevState[:] instead of state = deepcopy(prevState)
        Huge performance enhancement (~5-10x faster)
        Should be identical behavior if the items in the state list are immutable.
        (immutable objects include integers and strings so should be safe)
"""

# 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, math, sys, time
try:
    from numpy import random
except ImportError:
    import random

def round_figures(x, n):
    """Returns x rounded to n significant figures."""
    return round(x, int(n - math.ceil(math.log10(abs(x)))))

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)

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
    
    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:
                print(' Temperature        Energy    Accept   Improve     Elapsed   Remaining')
                print('%12.2f  %12.2f                      %s            ' % \
                    (T, E, time_string(elapsed) ) )
            else:
                remain = ( steps - step ) * ( elapsed / step )
                print('%12.2f  %12.2f  %7.2f%%  %7.2f%%  %s  %s' % \
                    (T, E, 100.0*acceptance, 100.0*improvement,
                        time_string(elapsed), time_string(remain)) )
        
        # Precompute factor for exponential cooling from Tmax to Tmin
        if Tmin <= 0.0:
            print('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 = copy.deepcopy(state)
        prevState = state[:]
        prevEnergy = E
        #bestState = copy.deepcopy(state)
        bestState = state[:]
        bestEnergy = E
        trials, accepts, improves = 0, 0, 0
        if updates > 0:
            updateWavelength = float(steps) / 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 = copy.deepcopy(prevState)
                state = prevState[:]
                E = prevEnergy
            else:
                # Accept new state and compare to best state
                accepts += 1
                if dE < 0.0:
                    improves += 1
                #prevState = copy.deepcopy(state)
                prevState = state[:]
                prevEnergy = E
                if E < bestEnergy:
                    #bestState = copy.deepcopy(state)
                    bestState = state[:]
                    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
    
    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 = copy.deepcopy(state)
            prevState = state[:]
            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 = copy.deepcopy(prevState)
                    state = prevState[:]
                    E = prevEnergy
                else:
                    accepts += 1
                    if dE < 0.0:
                        improves += 1
                    #prevState = copy.deepcopy(state)
                    prevState = state[:]
                    prevEnergy = E
            return state, E, float(accepts)/steps, float(improves)/steps
        
        step = 0
        start = time.time()
        
        print('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 )
        
        print('Exploring temperature landscape:')
        print(' 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
            print('%12.2f  %12.2f  %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)
        
        # MP: Don't perform anneal, just return params
        #return self.anneal(state, Tmax, Tmin, duration, 20)
        return {'tmax': Tmax, 'tmin': Tmin, 'steps': duration}

if __name__ == '__main__':
    """Test annealer with a traveling salesman problem."""
    
    # List latitude and longitude (degrees) for the twenty largest U.S. cities
    cities = { 'New York City': (40.72,74.00), 'Los Angeles': (34.05,118.25),
    'Chicago': (41.88,87.63), 'Houston': (29.77,95.38),
    'Phoenix': (33.45,112.07), 'Philadelphia': (39.95,75.17),
    'San Antonio': (29.53,98.47), 'Dallas': (32.78,96.80),
    'San Diego': (32.78,117.15), 'San Jose': (37.30,121.87),
    'Detroit': (42.33,83.05), 'San Francisco': (37.78,122.42),
    'Jacksonville': (30.32,81.70), 'Indianapolis': (39.78,86.15),
    'Austin': (30.27,97.77), 'Columbus': (39.98,82.98),
    'Fort Worth': (32.75,97.33), 'Charlotte': (35.23,80.85),
    'Memphis': (35.12,89.97), 'Baltimore': (39.28,76.62) }
    
    def distance(a, b):
        """Calculates distance between two latitude-longitude coordinates."""
        R = 3963  # radius of Earth (miles)
        lat1, lon1 = math.radians(a[0]), math.radians(a[1])
        lat2, lon2 = math.radians(b[0]), math.radians(b[1])
        return math.acos( math.sin(lat1)*math.sin(lat2) +
            math.cos(lat1)*math.cos(lat2)*math.cos(lon1-lon2) ) * R
    
    def route_move(state):
        """Swaps two cities in the route."""
        a = random.randint( 0, len(state)-1 )
        b = random.randint( 0, len(state)-1 )
        state[a], state[b] = state[b], state[a]
    
    def route_energy(state):
        """Calculates the length of the route."""
        e = 0
        for i in range(len(state)):
            e += distance( cities[state[i-1]], cities[state[i]] )
        return e
    
    # Start with the cities listed in random order
    randState = cities.keys()
    random.shuffle(randState)

    # Minimize the distance to be traveled by simulated annealing with a
    # manually chosen temperature schedule
    annealer = Annealer(route_energy, route_move)
    state, e = annealer.anneal(randState, 10000000, 0.01,
        18000 * len(randState), 9)
    while state[0] != 'New York City':
        state = state[1:] + state[:1]  # rotate NYC to start
    print("%i mile route:" % route_energy(state))
    for city in state:
        print("\t", city)
    
    # Minimize the distance to be traveled by simulated annealing with an
    # automatically chosen temperature schedule
    autoSchedule = annealer.auto(randState, 4)
    state, e = annealer.anneal(randState, autoSchedule['tmax'],
        autoSchedule['tmin'], autoSchedule['steps'],
	math.log(autoSchedule['tmax'] / autoSchedule['tmin']) / math.log(10))
    while state[0] != 'New York City':
        state = state[1:] + state[:1]  # rotate NYC to start
    print("%i mile route:" % route_energy(state))
    for city in state:
        print("\t", city)
    
    sys.exit()