def _convertBounds(bounds):
if bounds is None:
raise ValueError('bounds need to be defined')
if isinstance(bounds, Bounds):
limits = np.array(new_bounds_to_old(bounds.lb, bounds.ub,
len(bounds.lb)),
dtype=float).T
else:
limits = np.array(bounds, dtype='float').T
if (np.size(limits, 0) != 2 or not np.all(np.isfinite(limits))):
raise ValueError('bounds should be a sequence containing '
'real valued (min, max) pairs for each value'
' in x')
return limits[0], limits[1]
def test_new_bounds_to_old():
lb = np.array([-np.inf, 2, 3])
ub = np.array([3, np.inf, 10])
bounds = [(None, 3), (2, None), (3, 10)]
assert_array_equal(new_bounds_to_old(lb, ub, 3), bounds)
bounds_single_lb = [(-1, 3), (-1, None), (-1, 10)]
assert_array_equal(new_bounds_to_old(-1, ub, 3), bounds_single_lb)
bounds_no_lb = [(None, 3), (None, None), (None, 10)]
assert_array_equal(new_bounds_to_old(-np.inf, ub, 3), bounds_no_lb)
bounds_single_ub = [(None, 20), (2, 20), (3, 20)]
assert_array_equal(new_bounds_to_old(lb, 20, 3), bounds_single_ub)
bounds_no_ub = [(None, None), (2, None), (3, None)]
assert_array_equal(new_bounds_to_old(lb, np.inf, 3), bounds_no_ub)
bounds_single_both = [(1, 2), (1, 2), (1, 2)]
assert_array_equal(new_bounds_to_old(1, 2, 3), bounds_single_both)
bounds_no_both = [(None, None), (None, None), (None, None)]
assert_array_equal(new_bounds_to_old(-np.inf, np.inf, 3), bounds_no_both)
def test_new_bounds_to_old():
lb = np.array([-np.inf, 2, 3])
ub = np.array([3, np.inf, 10])
bounds = [(None, 3), (2, None), (3, 10)]
assert_array_equal(new_bounds_to_old(lb, ub, 3), bounds)
bounds_single_lb = [(-1, 3), (-1, None), (-1, 10)]
assert_array_equal(new_bounds_to_old(-1, ub, 3), bounds_single_lb)
bounds_no_lb = [(None, 3), (None, None), (None, 10)]
assert_array_equal(new_bounds_to_old(-np.inf, ub, 3), bounds_no_lb)
bounds_single_ub = [(None, 20), (2, 20), (3, 20)]
assert_array_equal(new_bounds_to_old(lb, 20, 3), bounds_single_ub)
bounds_no_ub = [(None, None), (2, None), (3, None)]
assert_array_equal(new_bounds_to_old(lb, np.inf, 3), bounds_no_ub)
bounds_single_both = [(1, 2), (1, 2), (1, 2)]
assert_array_equal(new_bounds_to_old(1, 2, 3), bounds_single_both)
bounds_no_both = [(None, None), (None, None), (None, None)]
assert_array_equal(new_bounds_to_old(-np.inf, np.inf, 3), bounds_no_both)
def dual_annealing(func,
bounds,
args=(),
maxiter=1000,
minimizer_kwargs=None,
initial_temp=5230.,
restart_temp_ratio=2.e-5,
visit=2.62,
accept=-5.0,
maxfun=1e7,
seed=None,
no_local_search=False,
callback=None,
x0=None,
local_search_options=None):
"""
Find the global minimum of a function using Dual Annealing.
Parameters
----------
func : callable
The objective function to be minimized. Must be in the form
``f(x, *args)``, where ``x`` is the argument in the form of a 1-D array
and ``args`` is a tuple of any additional fixed parameters needed to
completely specify the function.
bounds : sequence or `Bounds`
Bounds for variables. There are two ways to specify the bounds:
1. Instance of `Bounds` class.
2. Sequence of ``(min, max)`` pairs for each element in `x`.
args : tuple, optional
Any additional fixed parameters needed to completely specify the
objective function.
maxiter : int, optional
The maximum number of global search iterations. Default value is 1000.
minimizer_kwargs : dict, optional
Extra keyword arguments to be passed to the local minimizer
(`minimize`). Some important options could be:
``method`` for the minimizer method to use and ``args`` for
objective function additional arguments.
initial_temp : float, optional
The initial temperature, use higher values to facilitates a wider
search of the energy landscape, allowing dual_annealing to escape
local minima that it is trapped in. Default value is 5230. Range is
(0.01, 5.e4].
restart_temp_ratio : float, optional
During the annealing process, temperature is decreasing, when it
reaches ``initial_temp * restart_temp_ratio``, the reannealing process
is triggered. Default value of the ratio is 2e-5. Range is (0, 1).
visit : float, optional
Parameter for visiting distribution. Default value is 2.62. Higher
values give the visiting distribution a heavier tail, this makes
the algorithm jump to a more distant region. The value range is (1, 3].
accept : float, optional
Parameter for acceptance distribution. It is used to control the
probability of acceptance. The lower the acceptance parameter, the
smaller the probability of acceptance. Default value is -5.0 with
a range (-1e4, -5].
maxfun : int, optional
Soft limit for the number of objective function calls. If the
algorithm is in the middle of a local search, this number will be
exceeded, the algorithm will stop just after the local search is
done. Default value is 1e7.
seed : {None, int, `numpy.random.Generator`,
`numpy.random.RandomState`}, optional
If `seed` is None (or `np.random`), the `numpy.random.RandomState`
singleton is used.
If `seed` is an int, a new ``RandomState`` instance is used,
seeded with `seed`.
If `seed` is already a ``Generator`` or ``RandomState`` instance then
that instance is used.
Specify `seed` for repeatable minimizations. The random numbers
generated with this seed only affect the visiting distribution function
and new coordinates generation.
no_local_search : bool, optional
If `no_local_search` is set to True, a traditional Generalized
Simulated Annealing will be performed with no local search
strategy applied.
callback : callable, optional
A callback function with signature ``callback(x, f, context)``,
which will be called for all minima found.
``x`` and ``f`` are the coordinates and function value of the
latest minimum found, and ``context`` has value in [0, 1, 2], with the
following meaning:
- 0: minimum detected in the annealing process.
- 1: detection occurred in the local search process.
- 2: detection done in the dual annealing process.
If the callback implementation returns True, the algorithm will stop.
x0 : ndarray, shape(n,), optional
Coordinates of a single N-D starting point.
local_search_options : dict, optional
Backwards compatible flag for `minimizer_kwargs`, only one of these
should be supplied.
Returns
-------
res : OptimizeResult
The optimization result represented as a `OptimizeResult` object.
Important attributes are: ``x`` the solution array, ``fun`` the value
of the function at the solution, and ``message`` which describes the
cause of the termination.
See `OptimizeResult` for a description of other attributes.
Notes
-----
This function implements the Dual Annealing optimization. This stochastic
approach derived from [3]_ combines the generalization of CSA (Classical
Simulated Annealing) and FSA (Fast Simulated Annealing) [1]_ [2]_ coupled
to a strategy for applying a local search on accepted locations [4]_.
An alternative implementation of this same algorithm is described in [5]_
and benchmarks are presented in [6]_. This approach introduces an advanced
method to refine the solution found by the generalized annealing
process. This algorithm uses a distorted Cauchy-Lorentz visiting
distribution, with its shape controlled by the parameter :math:`q_{v}`
.. math::
g_{q_{v}}(\\Delta x(t)) \\propto \\frac{ \\
\\left[T_{q_{v}}(t) \\right]^{-\\frac{D}{3-q_{v}}}}{ \\
\\left[{1+(q_{v}-1)\\frac{(\\Delta x(t))^{2}} { \\
\\left[T_{q_{v}}(t)\\right]^{\\frac{2}{3-q_{v}}}}}\\right]^{ \\
\\frac{1}{q_{v}-1}+\\frac{D-1}{2}}}
Where :math:`t` is the artificial time. This visiting distribution is used
to generate a trial jump distance :math:`\\Delta x(t)` of variable
:math:`x(t)` under artificial temperature :math:`T_{q_{v}}(t)`.
From the starting point, after calling the visiting distribution
function, the acceptance probability is computed as follows:
.. math::
p_{q_{a}} = \\min{\\{1,\\left[1-(1-q_{a}) \\beta \\Delta E \\right]^{ \\
\\frac{1}{1-q_{a}}}\\}}
Where :math:`q_{a}` is a acceptance parameter. For :math:`q_{a}<1`, zero
acceptance probability is assigned to the cases where
.. math::
[1-(1-q_{a}) \\beta \\Delta E] < 0
The artificial temperature :math:`T_{q_{v}}(t)` is decreased according to
.. math::
T_{q_{v}}(t) = T_{q_{v}}(1) \\frac{2^{q_{v}-1}-1}{\\left( \\
1 + t\\right)^{q_{v}-1}-1}
Where :math:`q_{v}` is the visiting parameter.
.. versionadded:: 1.2.0
References
----------
.. [1] Tsallis C. Possible generalization of Boltzmann-Gibbs
statistics. Journal of Statistical Physics, 52, 479-487 (1998).
.. [2] Tsallis C, Stariolo DA. Generalized Simulated Annealing.
Physica A, 233, 395-406 (1996).
.. [3] Xiang Y, Sun DY, Fan W, Gong XG. Generalized Simulated
Annealing Algorithm and Its Application to the Thomson Model.
Physics Letters A, 233, 216-220 (1997).
.. [4] Xiang Y, Gong XG. Efficiency of Generalized Simulated
Annealing. Physical Review E, 62, 4473 (2000).
.. [5] Xiang Y, Gubian S, Suomela B, Hoeng J. Generalized
Simulated Annealing for Efficient Global Optimization: the GenSA
Package for R. The R Journal, Volume 5/1 (2013).
.. [6] Mullen, K. Continuous Global Optimization in R. Journal of
Statistical Software, 60(6), 1 - 45, (2014).
:doi:`10.18637/jss.v060.i06`
Examples
--------
The following example is a 10-D problem, with many local minima.
The function involved is called Rastrigin
(https://en.wikipedia.org/wiki/Rastrigin_function)
>>> from scipy.optimize import dual_annealing
>>> func = lambda x: np.sum(x*x - 10*np.cos(2*np.pi*x)) + 10*np.size(x)
>>> lw = [-5.12] * 10
>>> up = [5.12] * 10
>>> ret = dual_annealing(func, bounds=list(zip(lw, up)))
>>> ret.x
array([-4.26437714e-09, -3.91699361e-09, -1.86149218e-09, -3.97165720e-09,
-6.29151648e-09, -6.53145322e-09, -3.93616815e-09, -6.55623025e-09,
-6.05775280e-09, -5.00668935e-09]) # random
>>> ret.fun
0.000000
"""
if isinstance(bounds, Bounds):
bounds = new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb))
# noqa: E501
if x0 is not None and not len(x0) == len(bounds):
raise ValueError('Bounds size does not match x0')
lu = list(zip(*bounds))
lower = np.array(lu[0])
upper = np.array(lu[1])
# Check that restart temperature ratio is correct
if restart_temp_ratio <= 0. or restart_temp_ratio >= 1.:
raise ValueError('Restart temperature ratio has to be in range (0, 1)')
# Checking bounds are valid
if (np.any(np.isinf(lower)) or np.any(np.isinf(upper))
or np.any(np.isnan(lower)) or np.any(np.isnan(upper))):
raise ValueError('Some bounds values are inf values or nan values')
# Checking that bounds are consistent
if not np.all(lower < upper):
raise ValueError('Bounds are not consistent min < max')
# Checking that bounds are the same length
if not len(lower) == len(upper):
raise ValueError('Bounds do not have the same dimensions')
# Wrapper for the objective function
func_wrapper = ObjectiveFunWrapper(func, maxfun, *args)
# Wrapper for the minimizer
if local_search_options and minimizer_kwargs:
raise ValueError(
"dual_annealing only allows either 'minimizer_kwargs' (preferred) or "
"'local_search_options' (deprecated); not both!")
if local_search_options is not None:
warnings.warn(
"dual_annealing argument 'local_search_options' is "
"deprecated in favor of 'minimizer_kwargs'",
category=DeprecationWarning,
stacklevel=2)
minimizer_kwargs = local_search_options
# minimizer_kwargs has to be a dict, not None
minimizer_kwargs = minimizer_kwargs or {}
minimizer_wrapper = LocalSearchWrapper(bounds, func_wrapper,
**minimizer_kwargs)
# Initialization of random Generator for reproducible runs if seed provided
rand_state = check_random_state(seed)
# Initialization of the energy state
energy_state = EnergyState(lower, upper, callback)
energy_state.reset(func_wrapper, rand_state, x0)
# Minimum value of annealing temperature reached to perform
# re-annealing
temperature_restart = initial_temp * restart_temp_ratio
# VisitingDistribution instance
visit_dist = VisitingDistribution(lower, upper, visit, rand_state)
# Strategy chain instance
strategy_chain = StrategyChain(accept, visit_dist, func_wrapper,
minimizer_wrapper, rand_state, energy_state)
need_to_stop = False
iteration = 0
message = []
# OptimizeResult object to be returned
optimize_res = OptimizeResult()
optimize_res.success = True
optimize_res.status = 0
t1 = np.exp((visit - 1) * np.log(2.0)) - 1.0
# Run the search loop
while (not need_to_stop):
for i in range(maxiter):
# Compute temperature for this step
s = float(i) + 2.0
t2 = np.exp((visit - 1) * np.log(s)) - 1.0
temperature = initial_temp * t1 / t2
if iteration >= maxiter:
message.append("Maximum number of iteration reached")
need_to_stop = True
break
# Need a re-annealing process?
if temperature < temperature_restart:
energy_state.reset(func_wrapper, rand_state)
break
# starting strategy chain
val = strategy_chain.run(i, temperature)
if val is not None:
message.append(val)
need_to_stop = True
optimize_res.success = False
break
# Possible local search at the end of the strategy chain
if not no_local_search:
val = strategy_chain.local_search()
if val is not None:
message.append(val)
need_to_stop = True
optimize_res.success = False
break
iteration += 1
# Setting the OptimizeResult values
optimize_res.x = energy_state.xbest
optimize_res.fun = energy_state.ebest
optimize_res.nit = iteration
optimize_res.nfev = func_wrapper.nfev
optimize_res.njev = func_wrapper.ngev
optimize_res.nhev = func_wrapper.nhev
optimize_res.message = message
return optimize_res
def __init__(self, func, bounds, config, args=(), strategy='best1bin',
maxiter=1000, popsize=15, tol=0.01, mutation=(0.5, 1),
recombination=0.7, seed=None, maxfun=np.inf, callback=None,
disp=False, polish=True, init='latinhypercube', atol=0,
updating='immediate', workers=1):
self.config = config
if strategy in self._binomial:
self.mutation_func = getattr(self, self._binomial[strategy])
elif strategy in self._exponential:
self.mutation_func = getattr(self, self._exponential[strategy])
else:
raise ValueError("Please select a valid mutation strategy")
self.strategy = strategy
self.callback = callback
self.polish = polish
# set the updating / parallelisation options
if updating in ['immediate', 'deferred']:
self._updating = updating
# want to use parallelisation, but updating is immediate
if workers != 1 and updating == 'immediate':
warnings.warn("differential_evolution: the 'workers' keyword has"
" overridden updating='immediate' to"
" updating='deferred'", UserWarning)
self._updating = 'deferred'
# an object with a map method.
self._mapwrapper = MapWrapper(workers)
# relative and absolute tolerances for convergence
self.tol, self.atol = tol, atol
# Mutation constant should be in [0, 2). If specified as a sequence
# then dithering is performed.
self.scale = mutation
if (not np.all(np.isfinite(mutation)) or
np.any(np.array(mutation) >= 2) or
np.any(np.array(mutation) < 0)):
raise ValueError('The mutation constant must be a float in '
'U[0, 2), or specified as a tuple(min, max)'
' where min < max and min, max are in U[0, 2).')
self.dither = None
if hasattr(mutation, '__iter__') and len(mutation) > 1:
self.dither = [mutation[0], mutation[1]]
self.dither.sort()
self.cross_over_probability = recombination
# we create a wrapped function to allow the use of map (and Pool.map
# in the future)
self.func = _FunctionWrapper(func, args)
self.args = args
# convert tuple of lower and upper bounds to limits
# [(low_0, high_0), ..., (low_n, high_n]
# -> [[low_0, ..., low_n], [high_0, ..., high_n]]
if isinstance(bounds, Bounds):
self.limits = np.array(new_bounds_to_old(bounds.lb,
bounds.ub,
len(bounds.lb)),
dtype=float).T
else:
self.limits = np.array(bounds, dtype='float').T
if (np.size(self.limits, 0) != 2 or not
np.all(np.isfinite(self.limits))):
raise ValueError('bounds should be a sequence containing '
'real valued (min, max) pairs for each value'
' in x')
if maxiter is None: # the default used to be None
maxiter = 1000
self.maxiter = maxiter
if maxfun is None: # the default used to be None
maxfun = np.inf
self.maxfun = maxfun
# population is scaled to between [0, 1].
# We have to scale between parameter <-> population
# save these arguments for _scale_parameter and
# _unscale_parameter. This is an optimization
self.__scale_arg1 = 0.5 * (self.limits[0] + self.limits[1])
self.__scale_arg2 = np.fabs(self.limits[0] - self.limits[1])
self.parameter_count = np.size(self.limits, 1)
self.random_number_generator = check_random_state(seed)
# default population initialization is a latin hypercube design, but
# there are other population initializations possible.
# the minimum is 5 because 'best2bin' requires a population that's at
# least 5 long
self.num_population_members = max(5, popsize * self.parameter_count)
self.population_shape = (self.num_population_members,
self.parameter_count)
self._nfev = 0
if isinstance(init, string_types):
if init == 'latinhypercube':
self.init_population_lhs()
elif init == 'random':
self.init_population_random()
else:
raise ValueError(self.__init_error_msg)
else:
self.init_population_array(init)
self.disp = disp
self.fx_history = []
self.x_best_history = np.zeros((self.maxiter, self.parameter_count))