def _minimize_qpso(fun,
x0,
confunc=None,
g=.96,
max_iter=1000,
stable_iter=40,
ptol=1e-6,
ctol=1e-6,
levy_rate=0,
decay_rate=0,
reduction_rate=0.5,
callback=None,
verbose=False,
savefile=None):
"""Internal implementation for ``psopy.minimize_qpso``.
See Also
--------
psopy.minimize_qpso : The SciPy compatible interface to this function. Refer to
its documentation for an explanation of the parameters.
psopy.gen_confunc : Utility function to convert SciPy style constraints
to the form required by this function.
Parameters
----------
x0 : array_like of shape (N, D)
Initial position to begin QPSO from, where ``N`` is the number of points
and ``D`` the dimensionality of each point. For the constrained case
these points should satisfy all constraints.
fun : callable
The objective function to be minimized. Must be in the form
``fun(pos, *args)``. The argument ``pos``, is a 2-D array for initial
positions, where each row specifies the position of a different
particle, and ``args`` is a tuple of any additional fixed parameters
needed to completely specify the function.
confunc : callable
The function that describes constraints. Must be of the form
``confunc(pos)`` that returns the constraint matrix.
levy_rate: float
Whether to run the levy decay qpso or not. > 0 value turns on levy walk
decay_rate: float
Whether to turn on the decay function or not. > 0 value turns on the decay rate
Notes
-----
Chaotic Quantum PSO
Using this function directly allows for a slightly faster implementation
that does away with the need for the additional recursive calls needed to
wrap the constraint and objective functions for compatibility with Scipy.
"""
if verbose:
message = setup_print(x0.shape[1], max_iter, confunc is not None)
if savefile:
iterinfo = []
position = np.copy(x0)
nparam = len(position)
pbest = np.copy(position)
gbest = pbest[np.argmin(fun(pbest))]
oldfit = fun(gbest[None])[0]
stable_count = 0
dimension = len(position[0])
#simple levy walk. Make a decay function which will push particles around using stable_iter,max_iter is reached, pushing them away from pbest
beta = 3 / 2
sigma = (gamma(1 + beta) * sin(pi * beta / 2) / (gamma(
(1 + beta) / 2) * beta * 2**((beta - 1) / 2)))**(1 / beta)
decay = 1
stepsize = 1.0
for ii in range(max_iter):
mbest = np.sum(pbest, axis=0) / pbest.shape[0]
u = np.random.normal(0, 1, size=dimension) * sigma
v = np.random.normal(0, 1, size=dimension)
step = u / abs(v)**(1 / beta)
psi_1 = uniform(0, 1)
psi_2 = uniform(0, 1)
dv_g = psi_1 * gbest
if confunc is not None:
leaders = np.argmin(distance.cdist(position, pbest, 'sqeuclidean'),
axis=1)
dv_l = psi_2 * pbest[leaders]
else:
dv_l = psi_2 * pbest
P = (dv_g + dv_l) / (psi_1 + psi_2)
u = uniform(0, 1, nparam)
stepsize = 1.0
for i in range(0, nparam):
if levy_rate > 0:
stepsize = 0.01 * step * (1 /
(0.0000001 + position[i] - gbest[i]))
if decay_rate > 0:
decay = stepsize * 5 * (0.001)**(ii /
(max_iter * 0.05)) + 1
if uniform(0, 1) > 0.5:
position[i] = P[i] - mbest * np.log(1 / u[i]) * decay
else:
position[i] = P[i] + mbest * np.log(1 / u[i]) * decay
to_update = (fun(position) < fun(pbest))
if confunc is not None:
to_update &= (confunc(position).sum(axis=1) < ctol)
if to_update.any():
pbest[to_update] = position[to_update]
gbest = pbest[np.argmin(fun(pbest))]
# Termination criteria.
fval = fun(gbest[None])[0]
if np.abs(oldfit - fval) < ptol:
stable_count += 1
if stable_count == stable_iter:
break
else:
stable_count = 0
oldfit = fval
if verbose or savefile:
info = [ii, gbest, fval]
if confunc is not None:
cv = np.max(confunc(gbest[None]))
info.append(cv)
if verbose:
print(message.format(*info))
if savefile:
iterinfo.append(info)
# Final callback.
if callback is not None:
position = callback(position)
if savefile:
save_info(savefile, iterinfo, constraints=confunc is not None)
result = OptimizeResult(x=gbest,
fun=fun(gbest[None])[0],
nit=ii,
nsit=stable_count)
violation = False
if confunc is not None:
convec = confunc(gbest[None])
result.maxcv = np.max(convec)
result.cvec = convec
if convec.sum() > ctol:
violation = True
if violation:
result.status = 2
elif ii == max_iter:
result.status = 1
else:
result.status = 0
result.success = not result.status
return result
def _minimize_pso(fun,
x0,
confunc=None,
friction=.8,
max_velocity=5.,
g_rate=.8,
l_rate=.5,
max_iter=1000,
stable_iter=100,
ptol=1e-6,
ctol=1e-6,
callback=None,
verbose=False,
savefile=None):
"""Internal implementation for ``psopy.minimize``.
See Also
--------
psopy.minimize : The SciPy compatible interface to this function. Refer to
its documentation for an explanation of the parameters.
psopy.gen_confunc : Utility function to convert SciPy style constraints
to the form required by this function.
Parameters
----------
x0 : array_like of shape (N, D)
Initial position to begin PSO from, where ``N`` is the number of points
and ``D`` the dimensionality of each point. For the constrained case
these points should satisfy all constraints.
fun : callable
The objective function to be minimized. Must be in the form
``fun(pos, *args)``. The argument ``pos``, is a 2-D array for initial
positions, where each row specifies the position of a different
particle, and ``args`` is a tuple of any additional fixed parameters
needed to completely specify the function.
confunc : callable
The function that describes constraints. Must be of the form
``confunc(pos)`` that returns the constraint matrix.
Notes
-----
Using this function directly allows for a slightly faster implementation
that does away with the need for the additional recursive calls needed to
wrap the constraint and objective functions for compatibility with Scipy.
Examples
--------
These examples are identical to those laid out in ``psopy.minimize`` and
serve to illustrate the additional overhead in ensuring compatibility.
>>> import numpy as np
>>> from psopy import _minimize_pso
Consider the problem of minimizing the Rosenbrock function implemented as
``scipy.optimize.rosen``.
>>> from scipy.optimize import rosen
>>> fun = lambda x: np.apply_along_axis(rosen, 1, x)
Initialize 1000 particles and run the minimization function.
>>> x0 = np.random.uniform(0, 2, (1000, 5))
>>> res = _minimize_pso(fun, x0, stable_iter=50)
>>> res.x
array([1.00000003, 1.00000017, 1.00000034, 1.0000006 , 1.00000135])
Consider the constrained optimization problem with the objective function
defined as:
>>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2
>>> fun_ = lambda x: np.apply_along_axis(fun, 1, x)
and constraints defined as:
>>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2},
... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6},
... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2},
... {'type': 'ineq', 'fun': lambda x: x[0]},
... {'type': 'ineq', 'fun': lambda x: x[1]})
Initializing the constraint function and feasible solutions:
>>> from psopy import init_feasible, gen_confunc
>>> x0 = init_feasible(cons, low=0., high=2., shape=(1000, 2))
>>> confunc = gen_confunc(cons)
Running the constrained version of the function:
>>> res = _minimize_pso(fun_, x0, confunc=confunc, options={
... 'g_rate': 1., 'l_rate': 1., 'max_velocity': 4., 'stable_iter': 50})
>>> res.x
array([ 1.39985398, 1.69992748])
"""
if verbose:
message = setup_print(x0.shape[1], max_iter, confunc is not None)
if savefile:
iterinfo = []
position = np.copy(x0)
velocity = np.random.uniform(-max_velocity, max_velocity, position.shape)
pbest = np.copy(position)
gbest = pbest[np.argmin(fun(pbest))]
oldfit = fun(gbest[None])[0]
stable_count = 0
for ii in range(max_iter):
# Determine global and local gradient.
dv_g = g_rate * uniform(0, 1) * (gbest - position)
if confunc is not None:
leaders = np.argmin(distance.cdist(position, pbest, 'sqeuclidean'),
axis=1)
dv_l = l_rate * uniform(0, 1) * (pbest[leaders] - position)
else:
dv_l = l_rate * uniform(0, 1) * (pbest - position)
# Update velocity and position of particles.
velocity *= friction
velocity += (dv_g + dv_l)
np.clip(velocity, -max_velocity, max_velocity, out=velocity)
position += velocity
to_update = (fun(position) < fun(pbest))
if confunc is not None:
to_update &= (confunc(position).sum(axis=1) < ctol)
if to_update.any():
pbest[to_update] = position[to_update]
gbest = pbest[np.argmin(fun(pbest))]
# Termination criteria.
fval = fun(gbest[None])[0]
if np.abs(oldfit - fval) < ptol:
stable_count += 1
if stable_count == stable_iter:
break
else:
stable_count = 0
oldfit = fval
if verbose or savefile:
info = [ii, gbest, fval]
if confunc is not None:
cv = np.max(confunc(gbest[None]))
info.append(cv)
if verbose:
print(message.format(*info))
if savefile:
iterinfo.append(info)
# Final callback.
if callback is not None:
position = callback(position)
if savefile:
save_info(savefile, iterinfo, constraints=confunc is not None)
result = OptimizeResult(x=gbest,
fun=fun(gbest[None])[0],
nit=ii,
nsit=stable_count)
violation = False
if confunc is not None:
convec = confunc(gbest[None])
result.maxcv = np.max(convec)
result.cvec = convec
if convec.sum() > ctol:
violation = True
if violation:
result.status = 2
elif ii == max_iter:
result.status = 1
else:
result.status = 0
result.success = not result.status
return result
def optimize_minimize_mhmcmc_cluster(objective,
bounds,
args=(),
x0=None,
T=1,
N=3,
burnin=100000,
maxiter=1000000,
target_ar=0.4,
ar_tolerance=0.05,
cluster_eps=DEFAULT_CLUSTER_EPS,
rnd_seed=None,
collect_samples=None,
logger=None):
"""
Minimize objective function and return up to N local minima solutions.
:param objective: Objective function to minimize. Takes unpacked args as function call arguments and returns
a float.
:type objective: Callable(\*args) -> float
:param bounds: Bounds of the parameter space.
:type bounds: scipy.optimize.Bounds
:param args: Any additional fixed parameters needed to completely specify the objective function.
:type args: tuple or list
:param x0: Initial guess. If None, will be selected randomly and uniformly within the parameter bounds.
:type x0: numpy.array with same shape as elements of bounds
:param T: The "temperature" parameter for the accept or reject criterion. To sample the domain well,
should be in the order of the typical difference in local minima objective valuations.
:type T: float
:param N: Maximum number of minima to return
:type N: int
:param burnin: Number of random steps to discard before starting to accumulate statistics.
:type burnin: int
:param maxiter: Maximum number of steps to take (including burnin).
:type maxiter: int
:param target_ar: Target acceptance rate of point samples generated by stepping.
:type target_ar: float between 0 and 1
:param ar_tolerance: Tolerance on the acceptance rate before actively adapting the step size.
:type ar_tolerance: float
:param cluster_eps: Point proximity tolerance for DBSCAN clustering, in normalized bounds coordinates.
:type cluster_eps: float
:param rnd_seed: Random seed to force deterministic behaviour
:type rnd_seed: int
:param collect_samples: If not None and integral type, collect collect_samples at regular intervals
and return as part of solution.
:type collect_samples: int or NoneType
:param logger: Logger instance for outputting log messages.
:return: OptimizeResult containing solution(s) and solver data.
:rtype: scipy.optimize.OptimizeResult with additional attributes
"""
@call_counter
def obj_counted(*args):
return objective(*args)
# end func
assert maxiter >= 2 * burnin, "maxiter {} should be at least twice burnin steps {}".format(
maxiter, burnin)
main_iter = maxiter - burnin
if collect_samples is not None:
assert isinstance(collect_samples,
int), "collect_samples expected to be integral type"
assert collect_samples > 0, "collect_samples expected to be positive"
# end if
beta = 1.0 / T
if rnd_seed is None:
rnd_seed = int(time.time() * 1000) % (1 << 31)
# end if
np.random.seed(rnd_seed)
if logger:
logger.info('Using random seed {}'.format(rnd_seed))
# end
if x0 is None:
x0 = np.random.uniform(bounds.lb, bounds.ub)
# end if
assert np.all((x0 >= bounds.lb) & (x0 <= bounds.ub))
x = x0.copy()
funval = obj_counted(x, *args)
# Set up stepper with adaptive acceptance rate
stepper = BoundedRandNStepper(bounds)
stepper = AdaptiveStepsize(stepper,
accept_rate=target_ar,
ar_tolerance=ar_tolerance,
interval=50)
# -------------------------------
# DO BURN-IN
rejected_randomly = 0
accepted_burnin = 0
tracked_range = tqdm(range(burnin), total=burnin, desc='BURN-IN')
if logger:
stepper.logger = lambda msg: tracked_range.write(logger.name + ':' +
msg)
else:
stepper.logger = tracked_range.write
# end if
for _ in tracked_range:
x_new = stepper(x)
funval_new = obj_counted(x_new, *args)
log_alpha = -(funval_new - funval) * beta
if log_alpha > 0 or np.log(np.random.rand()) <= log_alpha:
x = x_new
funval = funval_new
stepper.notify_accept()
accepted_burnin += 1
elif log_alpha <= 0:
rejected_randomly += 1
# end if
# end for
ar = float(accepted_burnin) / burnin
if logger:
logger.info("Burn-in acceptance rate: {}".format(ar))
# end if
# -------------------------------
# DO MAIN LOOP
if collect_samples is not None:
nsamples = min(collect_samples, main_iter)
sample_cadence = main_iter / nsamples
samples = np.zeros((nsamples, len(x)))
samples_fval = np.zeros(nsamples)
# end if
accepted = 0
rejected_randomly = 0
minima_sorted = SortedList(
key=lambda rec: rec[1]) # Sort by objective function value
hist = HistogramIncremental(bounds, nbins=100)
# Cached a lot of potential minimum values, as these need to be clustered before return N results
N_cached = int(np.ceil(N * main_iter / 500))
next_sample = 0.0
sample_count = 0
tracked_range = tqdm(range(main_iter), total=main_iter, desc='MAIN')
if logger:
stepper.logger = lambda msg: tracked_range.write(logger.name + ':' +
msg)
else:
stepper.logger = tracked_range.write
# end if
for i in tracked_range:
if collect_samples and i >= next_sample:
assert sample_count < collect_samples
samples[sample_count] = x
samples_fval[sample_count] = funval
sample_count += 1
next_sample += sample_cadence
# end if
x_new = stepper(x)
funval_new = obj_counted(x_new, *args)
log_alpha = -(funval_new - funval) * beta
if log_alpha > 0 or np.log(np.random.rand()) <= log_alpha:
x = x_new
funval = funval_new
minima_sorted.add((x, funval))
if len(minima_sorted) > N_cached:
minima_sorted.pop()
# end if
stepper.notify_accept()
hist += x
accepted += 1
elif log_alpha <= 0:
rejected_randomly += 1
# end if
# end for
stepper.logger = None
ar = float(accepted) / main_iter
if logger:
logger.info("Acceptance rate: {}".format(ar))
logger.info("Best minima (before clustering):\n{}".format(
np.array([_mx[0] for _mx in minima_sorted[:10]])))
# end if
# -------------------------------
# Cluster minima and associate each cluster with a local minimum.
# Using a normalized coordinate space for cluster detection.
x_range = bounds.ub - bounds.lb
pts = np.array([x[0] for x in minima_sorted])
fvals = np.array([x[1] for x in minima_sorted])
pts_norm = (pts - bounds.lb) / x_range
_, labels = dbscan(pts_norm, eps=cluster_eps, min_samples=21, n_jobs=-1)
# Compute mean of each cluster and evaluate objective function at cluster mean locations.
minima_candidates = []
for grp in range(max(labels) + 1):
mask = (labels == grp)
mean_loc = np.mean(pts[mask, :], axis=0)
# Evaluate objective function precisely at the mean location of each cluster
fval = obj_counted(mean_loc, *args)
minima_candidates.append((mean_loc, grp, fval))
# end for
# Rank minima locations by objective function.
minima_candidates.sort(key=lambda c: c[2])
# Pick up to N solutions
solutions = minima_candidates[:N]
# Put results into OptimizeResult container.
# Add histograms to output result (in form of scipy.stats.rv_histogram)
solution = OptimizeResult()
solution.x = np.array([s[0] for s in solutions])
solution.clusters = [pts[(labels == s[1])] for s in solutions]
solution.cluster_funvals = [fvals[(labels == s[1])] for s in solutions]
solution.bins = hist.bins
solution.distribution = hist.histograms
solution.acceptance_rate = ar
solution.success = True
solution.status = 0
if len(solutions) > 0:
solution.message = 'SUCCESS: Found {} local minima'.format(
len(solutions))
else:
solution.message = 'WARNING: Found no clusters within tolerance {}'.format(
cluster_eps)
# end if
solution.fun = np.array([s[2] for s in solutions])
solution.jac = None
solution.nfev = obj_counted.counter
solution.njev = 0
solution.nit = main_iter
solution.maxcv = None
solution.samples = samples if collect_samples else None
solution.sample_funvals = samples_fval if collect_samples else None
solution.bounds = bounds
solution.version = 's0.3' # Solution version for future traceability
solution.rnd_seed = rnd_seed
return solution