Retracing a simulated-annealing's optimization steps - algorithm

Question

I'm using simulated annealing to help solve a problem such as the travelling salesman problem. I get a solution, which I'm happy with, but I would now like to know the path, within the solution space (that is, the steps taken by the algorithm to reach the solution), the algorithm took to go from a random, far from optimal itinerary, to pretty optimal one.
In other words, in the case of the traveling salesman problem:
Start with a random itinerary between cities
Step 1: path between city A-C and E-F were swapped
Step 2: path between city G-U and S-Q were swapped
...
End up with a pretty optimal itinerary between cities
Is there a way, using simulated annealing, to save each steps of the optimization process so that I can retrace, one by one, each change made to the system which eventually led to that particular solution? Or this goes against how a simulated annealing works?
Here is a great simulated annealing algorithm, by #perrygeo, very slightly modified from https://github.com/perrygeo/simanneal/blob/master/simanneal/anneal.py using the travelling salesman example: https://github.com/perrygeo/simanneal/blob/master/examples/salesman.py
(All my changes are followed by a one-line comment preceded by """)
from __future__ import division
from __future__ import print_function
from __future__ import absolute_import
from __future__ import unicode_literals
import copy
import math
import sys
import time
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(object):
"""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.
"""
Tmax = 25000.0
Tmin = 2.5
steps = 50000
updates = 100
copy_strategy = 'deepcopy'
def __init__(self, initial_state):
self.initial_state = initial_state
self.state = self.copy_state(initial_state)
def set_schedule(self, schedule):
"""Takes the output from `auto` and sets the attributes
"""
self.Tmax = schedule['tmax']
self.Tmin = schedule['tmin']
self.steps = int(schedule['steps'])
def copy_state(self, state):
"""Returns an exact copy of the provided state
Implemented according to self.copy_strategy, one of
* deepcopy : use copy.deepcopy (slow but reliable)
* slice: use list slices (faster but only works if state is list-like)
* method: use the state's copy() method
"""
if self.copy_strategy == 'deepcopy':
return copy.deepcopy(state)
elif self.copy_strategy == 'slice':
return state[:]
elif self.copy_strategy == 'method':
return state.copy()
def update(self, step, 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() - self.start
if step == 0:
print(' Temperature Energy Accept Improve Elapsed Remaining')
print('%12.2f %12.2f %s ' % \
(T, E, time_string(elapsed)))
else:
remain = (self.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)))
def anneal(self):
"""Minimizes the energy of a system by simulated annealing.
Parameters
state : an initial arrangement of the system
Returns
(state, energy): the best state and energy found.
"""
step = 0
self.start = time.time()
steps = [] ### initialise a list to save the steps taken by the algorithm to find a good solution
# Precompute factor for exponential cooling from Tmax to Tmin
if self.Tmin <= 0.0:
raise Exception('Exponential cooling requires a minimum "\
"temperature greater than zero.')
Tfactor = -math.log(self.Tmax / self.Tmin)
# Note initial state
T = self.Tmax
E = self.energy()
prevState = self.copy_state(self.state)
prevEnergy = E
bestState = self.copy_state(self.state)
bestEnergy = E
trials, accepts, improves = 0, 0, 0
if self.updates > 0:
updateWavelength = self.steps / self.updates
self.update(step, T, E, None, None)
# Attempt moves to new states
while step < self.steps:
step += 1
T = self.Tmax * math.exp(Tfactor * step / self.steps)
a,b = self.move()
E = self.energy()
dE = E - prevEnergy
trials += 1
if dE > 0.0 and math.exp(-dE / T) < random.random():
# Restore previous state
self.state = self.copy_state(prevState)
E = prevEnergy
else:
# Accept new state and compare to best state
accepts += 1
if dE < 0.0:
improves += 1
prevState = self.copy_state(self.state)
prevEnergy = E
steps.append([a,b]) ### append the "good move" to the list of steps
if E < bestEnergy:
bestState = self.copy_state(self.state)
bestEnergy = E
if self.updates > 1:
if step // updateWavelength > (step - 1) // updateWavelength:
self.update(
step, T, E, accepts / trials, improves / trials)
trials, accepts, improves = 0, 0, 0
# Return best state and energy
return bestState, bestEnergy, steps ### added steps to what should be returned
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
class TravellingSalesmanProblem(Annealer):
"""Test annealer with a travelling salesman problem.
"""
# pass extra data (the distance matrix) into the constructor
def __init__(self, state, distance_matrix):
self.distance_matrix = distance_matrix
super(TravellingSalesmanProblem, self).__init__(state) # important!
def move(self):
"""Swaps two cities in the route."""
a = random.randint(0, len(self.state) - 1)
b = random.randint(0, len(self.state) - 1)
self.state[a], self.state[b] = self.state[b], self.state[a]
return a,b ### return the change made
def energy(self):
"""Calculates the length of the route."""
e = 0
for i in range(len(self.state)):
e += self.distance_matrix[self.state[i-1]][self.state[i]]
return e
if __name__ == '__main__':
# latitude and longitude 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)
}
# initial state, a randomly-ordered itinerary
init_state = list(cities.keys())
random.shuffle(init_state)
reconstructed_state = init_state
# create a distance matrix
distance_matrix = {}
for ka, va in cities.items():
distance_matrix[ka] = {}
for kb, vb in cities.items():
if kb == ka:
distance_matrix[ka][kb] = 0.0
else:
distance_matrix[ka][kb] = distance(va, vb)
tsp = TravellingSalesmanProblem(init_state, distance_matrix)
# since our state is just a list, slice is the fastest way to copy
tsp.copy_strategy = "slice"
state, e, steps = tsp.anneal()
while state[0] != 'New York City':
state = state[1:] + state[:1] # rotate NYC to start
print("Results:")
for city in state:
print("\t", city)
### recontructed the annealing process
print("")
print("nbr. of steps:",len(steps))
print("Reconstructed results:")
for s in steps:
reconstructed_state[s[0]], reconstructed_state[s[1]] = reconstructed_state[s[1]], reconstructed_state[s[0]]
while reconstructed_state[0] != 'New York City':
reconstructed_state = reconstructed_state[1:] + reconstructed_state[:1] # rotate NYC to start
for city in reconstructed_state:
print("\t", city)
Saving every time a move is made builds a huge list of steps that are indeed retraceable. However it obviously mimics how the algorithm explores and jumps around many different positions in the solution space, especially at high temperatures.
To get more straightly converging steps, I could move step-saving line:
steps.append([a,b]) ### append the "good move" to the list of steps
under
if E < bestEnergy:
Where only the steps actually improving upon the best found solution so far are saved. However, the final list of steps does not help reconstruct the itinerary anymore (steps are missing).
Is this problem hopeless and inherent to how simulated annealing works, or is there hope to being able to construct a converging step-list from random to quasi-optimal?

Answer 1

Saving the steps along the way does not depend on the algorithm, i.e. simulated annealing, but on your implementation or software you use. If it is your own implementation, it should be no problem at all to save these steps. If you use an event-listener approach, you can add arbitrary no. of listeners/clients to process your events. But you can also just write out your data to one client, e.g. file. If you do not use your own implementation, it depends on the software. If you use proprietary software, you are dependent on its API. If you use open source software and you do not find ways to retrieve such information in its API, you are usually allowed to modify the software for your own needs.

Answer 2

Ultimately the move itself is not important except in the context of rest of the system state. Instead of trying to track the move or change in state that resulted in each low energy solution (which, as you point out, loses all the information about previous moves that were accepted), you could just track the state itself every time you encountered a new lowest score:
if E < bestEnergy:
list_of_best_states.append(self.copy_state(self.state))

Answer 3

Given any randomized algorithm, if you use a pseudorandom generator, then you can save the generator's seed (random but short) and replay the execution by seeding the generator the same way.

Answer 4

I have started with the Ingber ASA code for a more conventional SA problem involving estimating about 72 error variables to give a best fit to measured data.
In my case, there are a limited number of trials that result in a reduced cost. Of several million trials, typically less than 40 are accepted. If I wanted to keep track of them, I would add a step in the code that accepts a new position to log the difference between the then-current and the now-new position.
Replaying these differences would give you the path.