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Inzynierka/Lib/site-packages/scipy/optimize/_shgo.py

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2023-06-02 12:51:02 +02:00
"""
shgo: The simplicial homology global optimisation algorithm
"""
import numpy as np
import time
import logging
import warnings
from scipy import spatial
from scipy.optimize import OptimizeResult, minimize, Bounds
from scipy.optimize._constraints import new_bounds_to_old
from ._optimize import _wrap_scalar_function
from scipy.optimize._shgo_lib.triangulation import Complex
__all__ = ['shgo']
def shgo(func, bounds, args=(), constraints=None, n=None, iters=1,
callback=None,
minimizer_kwargs=None, options=None, sampling_method='simplicial'):
"""
Finds the global minimum of a function using SHG optimization.
SHGO stands for "simplicial homology global optimization".
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.
constraints : dict or sequence of dict, optional
Constraints definition.
Function(s) ``R**n`` in the form::
g(x) >= 0 applied as g : R^n -> R^m
h(x) == 0 applied as h : R^n -> R^p
Each constraint is defined in a dictionary with fields:
type : str
Constraint type: 'eq' for equality, 'ineq' for inequality.
fun : callable
The function defining the constraint.
jac : callable, optional
The Jacobian of `fun` (only for SLSQP).
args : sequence, optional
Extra arguments to be passed to the function and Jacobian.
Equality constraint means that the constraint function result is to
be zero whereas inequality means that it is to be non-negative.
Note that COBYLA only supports inequality constraints.
.. note::
Only the COBYLA and SLSQP local minimize methods currently
support constraint arguments. If the ``constraints`` sequence
used in the local optimization problem is not defined in
``minimizer_kwargs`` and a constrained method is used then the
global ``constraints`` will be used.
(Defining a ``constraints`` sequence in ``minimizer_kwargs``
means that ``constraints`` will not be added so if equality
constraints and so forth need to be added then the inequality
functions in ``constraints`` need to be added to
``minimizer_kwargs`` too).
n : int, optional
Number of sampling points used in the construction of the simplicial
complex. Note that this argument is only used for ``sobol`` and other
arbitrary `sampling_methods`. In case of ``sobol``, it must be a
power of 2: ``n=2**m``, and the argument will automatically be
converted to the next higher power of 2. Default is 100 for
``sampling_method='simplicial'`` and 128 for
``sampling_method='sobol'``.
iters : int, optional
Number of iterations used in the construction of the simplicial
complex. Default is 1.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
minimizer_kwargs : dict, optional
Extra keyword arguments to be passed to the minimizer
``scipy.optimize.minimize`` Some important options could be:
* method : str
The minimization method, the default is ``SLSQP``.
* args : tuple
Extra arguments passed to the objective function (``func``) and
its derivatives (Jacobian, Hessian).
* options : dict, optional
Note that by default the tolerance is specified as
``{ftol: 1e-12}``
options : dict, optional
A dictionary of solver options. Many of the options specified for the
global routine are also passed to the scipy.optimize.minimize routine.
The options that are also passed to the local routine are marked with
"(L)".
Stopping criteria, the algorithm will terminate if any of the specified
criteria are met. However, the default algorithm does not require any to
be specified:
* maxfev : int (L)
Maximum number of function evaluations in the feasible domain.
(Note only methods that support this option will terminate
the routine at precisely exact specified value. Otherwise the
criterion will only terminate during a global iteration)
* f_min
Specify the minimum objective function value, if it is known.
* f_tol : float
Precision goal for the value of f in the stopping
criterion. Note that the global routine will also
terminate if a sampling point in the global routine is
within this tolerance.
* maxiter : int
Maximum number of iterations to perform.
* maxev : int
Maximum number of sampling evaluations to perform (includes
searching in infeasible points).
* maxtime : float
Maximum processing runtime allowed
* minhgrd : int
Minimum homology group rank differential. The homology group of the
objective function is calculated (approximately) during every
iteration. The rank of this group has a one-to-one correspondence
with the number of locally convex subdomains in the objective
function (after adequate sampling points each of these subdomains
contain a unique global minimum). If the difference in the hgr is 0
between iterations for ``maxhgrd`` specified iterations the
algorithm will terminate.
Objective function knowledge:
* symmetry : bool
Specify True if the objective function contains symmetric variables.
The search space (and therefore performance) is decreased by O(n!).
* jac : bool or callable, optional
Jacobian (gradient) of objective function. Only for CG, BFGS,
Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg. If ``jac`` is a
boolean and is True, ``fun`` is assumed to return the gradient along
with the objective function. If False, the gradient will be
estimated numerically. ``jac`` can also be a callable returning the
gradient of the objective. In this case, it must accept the same
arguments as ``fun``. (Passed to `scipy.optimize.minmize` automatically)
* hess, hessp : callable, optional
Hessian (matrix of second-order derivatives) of objective function
or Hessian of objective function times an arbitrary vector p.
Only for Newton-CG, dogleg, trust-ncg. Only one of ``hessp`` or
``hess`` needs to be given. If ``hess`` is provided, then
``hessp`` will be ignored. If neither ``hess`` nor ``hessp`` is
provided, then the Hessian product will be approximated using
finite differences on ``jac``. ``hessp`` must compute the Hessian
times an arbitrary vector. (Passed to `scipy.optimize.minmize`
automatically)
Algorithm settings:
* minimize_every_iter : bool
If True then promising global sampling points will be passed to a
local minimization routine every iteration. If False then only the
final minimizer pool will be run. Defaults to False.
* local_iter : int
Only evaluate a few of the best minimizer pool candidates every
iteration. If False all potential points are passed to the local
minimization routine.
* infty_constraints: bool
If True then any sampling points generated which are outside will
the feasible domain will be saved and given an objective function
value of ``inf``. If False then these points will be discarded.
Using this functionality could lead to higher performance with
respect to function evaluations before the global minimum is found,
specifying False will use less memory at the cost of a slight
decrease in performance. Defaults to True.
Feedback:
* disp : bool (L)
Set to True to print convergence messages.
sampling_method : str or function, optional
Current built in sampling method options are ``halton``, ``sobol`` and
``simplicial``. The default ``simplicial`` provides
the theoretical guarantee of convergence to the global minimum in finite
time. ``halton`` and ``sobol`` method are faster in terms of sampling
point generation at the cost of the loss of
guaranteed convergence. It is more appropriate for most "easier"
problems where the convergence is relatively fast.
User defined sampling functions must accept two arguments of ``n``
sampling points of dimension ``dim`` per call and output an array of
sampling points with shape `n x dim`.
Returns
-------
res : OptimizeResult
The optimization result represented as a `OptimizeResult` object.
Important attributes are:
``x`` the solution array corresponding to the global minimum,
``fun`` the function output at the global solution,
``xl`` an ordered list of local minima solutions,
``funl`` the function output at the corresponding local solutions,
``success`` a Boolean flag indicating if the optimizer exited
successfully,
``message`` which describes the cause of the termination,
``nfev`` the total number of objective function evaluations including
the sampling calls,
``nlfev`` the total number of objective function evaluations
culminating from all local search optimizations,
``nit`` number of iterations performed by the global routine.
Notes
-----
Global optimization using simplicial homology global optimization [1]_.
Appropriate for solving general purpose NLP and blackbox optimization
problems to global optimality (low-dimensional problems).
In general, the optimization problems are of the form::
minimize f(x) subject to
g_i(x) >= 0, i = 1,...,m
h_j(x) = 0, j = 1,...,p
where x is a vector of one or more variables. ``f(x)`` is the objective
function ``R^n -> R``, ``g_i(x)`` are the inequality constraints, and
``h_j(x)`` are the equality constraints.
Optionally, the lower and upper bounds for each element in x can also be
specified using the `bounds` argument.
While most of the theoretical advantages of SHGO are only proven for when
``f(x)`` is a Lipschitz smooth function, the algorithm is also proven to
converge to the global optimum for the more general case where ``f(x)`` is
non-continuous, non-convex and non-smooth, if the default sampling method
is used [1]_.
The local search method may be specified using the ``minimizer_kwargs``
parameter which is passed on to ``scipy.optimize.minimize``. By default,
the ``SLSQP`` method is used. In general, it is recommended to use the
``SLSQP`` or ``COBYLA`` local minimization if inequality constraints
are defined for the problem since the other methods do not use constraints.
The ``halton`` and ``sobol`` method points are generated using
`scipy.stats.qmc`. Any other QMC method could be used.
References
----------
.. [1] Endres, SC, Sandrock, C, Focke, WW (2018) "A simplicial homology
algorithm for lipschitz optimisation", Journal of Global Optimization.
.. [2] Joe, SW and Kuo, FY (2008) "Constructing Sobol' sequences with
better two-dimensional projections", SIAM J. Sci. Comput. 30,
2635-2654.
.. [3] Hoch, W and Schittkowski, K (1981) "Test examples for nonlinear
programming codes", Lecture Notes in Economics and Mathematical
Systems, 187. Springer-Verlag, New York.
http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf
.. [4] Wales, DJ (2015) "Perspective: Insight into reaction coordinates and
dynamics from the potential energy landscape",
Journal of Chemical Physics, 142(13), 2015.
Examples
--------
First consider the problem of minimizing the Rosenbrock function, `rosen`:
>>> import numpy as np
>>> from scipy.optimize import rosen, shgo
>>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
>>> result = shgo(rosen, bounds)
>>> result.x, result.fun
(array([1., 1., 1., 1., 1.]), 2.920392374190081e-18)
Note that bounds determine the dimensionality of the objective
function and is therefore a required input, however you can specify
empty bounds using ``None`` or objects like ``np.inf`` which will be
converted to large float numbers.
>>> bounds = [(None, None), ]*4
>>> result = shgo(rosen, bounds)
>>> result.x
array([0.99999851, 0.99999704, 0.99999411, 0.9999882 ])
Next, we consider the Eggholder function, a problem with several local
minima and one global minimum. We will demonstrate the use of arguments and
the capabilities of `shgo`.
(https://en.wikipedia.org/wiki/Test_functions_for_optimization)
>>> def eggholder(x):
... return (-(x[1] + 47.0)
... * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0))))
... - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0))))
... )
...
>>> bounds = [(-512, 512), (-512, 512)]
`shgo` has built-in low discrepancy sampling sequences. First, we will
input 64 initial sampling points of the *Sobol'* sequence:
>>> result = shgo(eggholder, bounds, n=64, sampling_method='sobol')
>>> result.x, result.fun
(array([512. , 404.23180824]), -959.6406627208397)
`shgo` also has a return for any other local minima that was found, these
can be called using:
>>> result.xl
array([[ 512. , 404.23180824],
[ 283.0759062 , -487.12565635],
[-294.66820039, -462.01964031],
[-105.87688911, 423.15323845],
[-242.97926 , 274.38030925],
[-506.25823477, 6.3131022 ],
[-408.71980731, -156.10116949],
[ 150.23207937, 301.31376595],
[ 91.00920901, -391.283763 ],
[ 202.89662724, -269.38043241],
[ 361.66623976, -106.96493868],
[-219.40612786, -244.06020508]])
>>> result.funl
array([-959.64066272, -718.16745962, -704.80659592, -565.99778097,
-559.78685655, -557.36868733, -507.87385942, -493.9605115 ,
-426.48799655, -421.15571437, -419.31194957, -410.98477763])
These results are useful in applications where there are many global minima
and the values of other global minima are desired or where the local minima
can provide insight into the system (for example morphologies
in physical chemistry [4]_).
If we want to find a larger number of local minima, we can increase the
number of sampling points or the number of iterations. We'll increase the
number of sampling points to 64 and the number of iterations from the
default of 1 to 3. Using ``simplicial`` this would have given us
64 x 3 = 192 initial sampling points.
>>> result_2 = shgo(eggholder, bounds, n=64, iters=3, sampling_method='sobol')
>>> len(result.xl), len(result_2.xl)
(12, 20)
Note the difference between, e.g., ``n=192, iters=1`` and ``n=64,
iters=3``.
In the first case the promising points contained in the minimiser pool
are processed only once. In the latter case it is processed every 64
sampling points for a total of 3 times.
To demonstrate solving problems with non-linear constraints consider the
following example from Hock and Schittkowski problem 73 (cattle-feed) [3]_::
minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4
subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5 >= 0,
12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21
-1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 +
20.5 * x_3**2 + 0.62 * x_4**2) >= 0,
x_1 + x_2 + x_3 + x_4 - 1 == 0,
1 >= x_i >= 0 for all i
The approximate answer given in [3]_ is::
f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378
>>> def f(x): # (cattle-feed)
... return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3]
...
>>> def g1(x):
... return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5 # >=0
...
>>> def g2(x):
... return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21
... - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2
... + 20.5*x[2]**2 + 0.62*x[3]**2)
... ) # >=0
...
>>> def h1(x):
... return x[0] + x[1] + x[2] + x[3] - 1 # == 0
...
>>> cons = ({'type': 'ineq', 'fun': g1},
... {'type': 'ineq', 'fun': g2},
... {'type': 'eq', 'fun': h1})
>>> bounds = [(0, 1.0),]*4
>>> res = shgo(f, bounds, iters=3, constraints=cons)
>>> res
message: Optimization terminated successfully.
success: True
fun: 29.894378159142136
funl: [ 2.989e+01]
x: [ 6.355e-01 1.137e-13 3.127e-01 5.178e-02]
xl: [[ 6.355e-01 1.137e-13 3.127e-01 5.178e-02]]
nit: 3
nfev: 114
nlfev: 35
nljev: 5
nlhev: 0
>>> g1(res.x), g2(res.x), h1(res.x)
(-5.062616992290714e-14, -2.9594104944408173e-12, 0.0)
"""
# if necessary, convert bounds class to old bounds
if isinstance(bounds, Bounds):
bounds = new_bounds_to_old(bounds.lb, bounds.ub, len(bounds.lb))
# Initiate SHGO class
shc = SHGO(func, bounds, args=args, constraints=constraints, n=n,
iters=iters, callback=callback,
minimizer_kwargs=minimizer_kwargs,
options=options, sampling_method=sampling_method)
# Run the algorithm, process results and test success
shc.construct_complex()
if not shc.break_routine:
if shc.disp:
print("Successfully completed construction of complex.")
# Test post iterations success
if len(shc.LMC.xl_maps) == 0:
# If sampling failed to find pool, return lowest sampled point
# with a warning
shc.find_lowest_vertex()
shc.break_routine = True
shc.fail_routine(mes="Failed to find a feasible minimizer point. "
"Lowest sampling point = {}".format(shc.f_lowest))
shc.res.fun = shc.f_lowest
shc.res.x = shc.x_lowest
shc.res.nfev = shc.fn
# Confirm the routine ran successfully
if not shc.break_routine:
shc.res.message = 'Optimization terminated successfully.'
shc.res.success = True
# Return the final results
return shc.res
class SHGO:
def __init__(self, func, bounds, args=(), constraints=None, n=None,
iters=None, callback=None, minimizer_kwargs=None,
options=None, sampling_method='sobol'):
from scipy.stats import qmc
# Input checks
methods = ['halton', 'sobol', 'simplicial']
if isinstance(sampling_method, str) and sampling_method not in methods:
raise ValueError(("Unknown sampling_method specified."
" Valid methods: {}").format(', '.join(methods)))
# Initiate class
self._raw_func = func # some methods pass args in (e.g. Complex)
_, self.func = _wrap_scalar_function(func, args)
self.bounds = bounds
self.args = args
self.callback = callback
# Bounds
abound = np.array(bounds, float)
self.dim = np.shape(abound)[0] # Dimensionality of problem
# Set none finite values to large floats
infind = ~np.isfinite(abound)
abound[infind[:, 0], 0] = -1e50
abound[infind[:, 1], 1] = 1e50
# Check if bounds are correctly specified
bnderr = abound[:, 0] > abound[:, 1]
if bnderr.any():
raise ValueError('Error: lb > ub in bounds {}.'
.format(', '.join(str(b) for b in bnderr)))
self.bounds = abound
# Constraints
# Process constraint dict sequence:
if constraints is not None:
self.min_cons = constraints
self.g_cons = []
self.g_args = []
if (type(constraints) is not tuple) and (type(constraints)
is not list):
constraints = (constraints,)
for cons in constraints:
if cons['type'] == 'ineq':
self.g_cons.append(cons['fun'])
try:
self.g_args.append(cons['args'])
except KeyError:
self.g_args.append(())
self.g_cons = tuple(self.g_cons)
self.g_args = tuple(self.g_args)
else:
self.g_cons = None
self.g_args = None
# Define local minimization keyword arguments
# Start with defaults
self.minimizer_kwargs = {'method': 'SLSQP',
'bounds': self.bounds,
'options': {},
'callback': self.callback
}
if minimizer_kwargs is not None:
# Overwrite with supplied values
self.minimizer_kwargs.update(minimizer_kwargs)
else:
self.minimizer_kwargs['options'] = {'ftol': 1e-12}
if (self.minimizer_kwargs['method'] in ('SLSQP', 'COBYLA') and
(minimizer_kwargs is not None and
'constraints' not in minimizer_kwargs and
constraints is not None) or
(self.g_cons is not None)):
self.minimizer_kwargs['constraints'] = self.min_cons
# Process options dict
if options is not None:
self.init_options(options)
else: # Default settings:
self.f_min_true = None
self.minimize_every_iter = False
# Algorithm limits
self.maxiter = None
self.maxfev = None
self.maxev = None
self.maxtime = None
self.f_min_true = None
self.minhgrd = None
# Objective function knowledge
self.symmetry = False
# Algorithm functionality
self.local_iter = False
self.infty_cons_sampl = True
# Feedback
self.disp = False
# Remove unknown arguments in self.minimizer_kwargs
# Start with arguments all the solvers have in common
self.min_solver_args = ['fun', 'x0', 'args',
'callback', 'options', 'method']
# then add the ones unique to specific solvers
solver_args = {
'_custom': ['jac', 'hess', 'hessp', 'bounds', 'constraints'],
'nelder-mead': [],
'powell': [],
'cg': ['jac'],
'bfgs': ['jac'],
'newton-cg': ['jac', 'hess', 'hessp'],
'l-bfgs-b': ['jac', 'bounds'],
'tnc': ['jac', 'bounds'],
'cobyla': ['constraints'],
'slsqp': ['jac', 'bounds', 'constraints'],
'dogleg': ['jac', 'hess'],
'trust-ncg': ['jac', 'hess', 'hessp'],
'trust-krylov': ['jac', 'hess', 'hessp'],
'trust-exact': ['jac', 'hess'],
'trust-constr': ['jac', 'hess', 'hessp'],
}
method = self.minimizer_kwargs['method']
self.min_solver_args += solver_args[method.lower()]
# Only retain the known arguments
def _restrict_to_keys(dictionary, goodkeys):
"""Remove keys from dictionary if not in goodkeys - inplace"""
existingkeys = set(dictionary)
for key in existingkeys - set(goodkeys):
dictionary.pop(key, None)
_restrict_to_keys(self.minimizer_kwargs, self.min_solver_args)
_restrict_to_keys(self.minimizer_kwargs['options'],
self.min_solver_args + ['ftol'])
# Algorithm controls
# Global controls
self.stop_global = False # Used in the stopping_criteria method
self.break_routine = False # Break the algorithm globally
self.iters = iters # Iterations to be ran
self.iters_done = 0 # Iterations to be ran
self.n = n # Sampling points per iteration
self.nc = n # Sampling points to sample in current iteration
self.n_prc = 0 # Processed points (used to track Delaunay iters)
self.n_sampled = 0 # To track number of sampling points already generated
self.fn = 0 # Number of feasible sampling points evaluations performed
self.hgr = 0 # Homology group rank
# Default settings if no sampling criteria.
if self.iters is None:
self.iters = 1
if self.n is None:
self.n = 100
if sampling_method == 'sobol':
self.n = 128
self.nc = self.n
if not ((self.maxiter is None) and (self.maxfev is None) and (
self.maxev is None)
and (self.minhgrd is None) and (self.f_min_true is None)):
self.iters = None
# Set complex construction mode based on a provided stopping criteria:
# Choose complex constructor
if sampling_method == 'simplicial':
self.iterate_complex = self.iterate_hypercube
self.minimizers = self.simplex_minimizers
self.sampling_method = sampling_method
elif sampling_method in ['halton', 'sobol'] or \
not isinstance(sampling_method, str):
self.iterate_complex = self.iterate_delaunay
self.minimizers = self.delaunay_complex_minimisers
# Sampling method used
if sampling_method in ['halton', 'sobol']:
if sampling_method == 'sobol':
self.n = int(2 ** np.ceil(np.log2(self.n)))
self.nc = self.n
self.sampling_method = 'sobol'
self.qmc_engine = qmc.Sobol(d=self.dim, scramble=False,
seed=np.random.RandomState())
else:
self.sampling_method = 'halton'
self.qmc_engine = qmc.Halton(d=self.dim, scramble=True,
seed=np.random.RandomState())
sampling_method = lambda n, d: self.qmc_engine.random(n)
else:
# A user defined sampling method:
self.sampling_method = 'custom'
self.sampling = self.sampling_custom
self.sampling_function = sampling_method # F(n, d)
# Local controls
self.stop_l_iter = False # Local minimisation iterations
self.stop_complex_iter = False # Sampling iterations
# Initiate storage objects used in algorithm classes
self.minimizer_pool = []
# Cache of local minimizers mapped
self.LMC = LMapCache()
# Initialize return object
self.res = OptimizeResult() # scipy.optimize.OptimizeResult object
self.res.nfev = 0 # Includes each sampling point as func evaluation
self.res.nlfev = 0 # Local function evals for all minimisers
self.res.nljev = 0 # Local Jacobian evals for all minimisers
self.res.nlhev = 0 # Local Hessian evals for all minimisers
# Initiation aids
def init_options(self, options):
"""
Initiates the options.
Can also be useful to change parameters after class initiation.
Parameters
----------
options : dict
Returns
-------
None
"""
# Update 'options' dict passed to optimize.minimize
# Do this first so we don't mutate `options` below.
self.minimizer_kwargs['options'].update(options)
# Ensure that 'jac', 'hess', and 'hessp' are passed directly to
# `minimize` as keywords, not as part of its 'options' dictionary.
for opt in ['jac', 'hess', 'hessp']:
if opt in self.minimizer_kwargs['options']:
self.minimizer_kwargs[opt] = (
self.minimizer_kwargs['options'].pop(opt))
# Default settings:
self.minimize_every_iter = options.get('minimize_every_iter', False)
# Algorithm limits
# Maximum number of iterations to perform.
self.maxiter = options.get('maxiter', None)
# Maximum number of function evaluations in the feasible domain
self.maxfev = options.get('maxfev', None)
# Maximum number of sampling evaluations (includes searching in
# infeasible points
self.maxev = options.get('maxev', None)
# Maximum processing runtime allowed
self.init = time.time()
self.maxtime = options.get('maxtime', None)
if 'f_min' in options:
# Specify the minimum objective function value, if it is known.
self.f_min_true = options['f_min']
self.f_tol = options.get('f_tol', 1e-4)
else:
self.f_min_true = None
self.minhgrd = options.get('minhgrd', None)
# Objective function knowledge
self.symmetry = 'symmetry' in options
# Algorithm functionality
# Only evaluate a few of the best candiates
self.local_iter = options.get('local_iter', False)
self.infty_cons_sampl = options.get('infty_constraints', True)
# Feedback
self.disp = options.get('disp', False)
# Iteration properties
# Main construction loop:
def construct_complex(self):
"""
Construct for `iters` iterations.
If uniform sampling is used, every iteration adds 'n' sampling points.
Iterations if a stopping criteria (e.g., sampling points or
processing time) has been met.
"""
if self.disp:
print('Splitting first generation')
while not self.stop_global:
if self.break_routine:
break
# Iterate complex, process minimisers
self.iterate()
self.stopping_criteria()
# Build minimiser pool
# Final iteration only needed if pools weren't minimised every iteration
if not self.minimize_every_iter:
if not self.break_routine:
self.find_minima()
self.res.nit = self.iters_done + 1
def find_minima(self):
"""
Construct the minimizer pool, map the minimizers to local minima
and sort the results into a global return object.
"""
self.minimizers()
if len(self.X_min) != 0:
# Minimize the pool of minimizers with local minimization methods
# Note that if Options['local_iter'] is an `int` instead of default
# value False then only that number of candidates will be minimized
self.minimise_pool(self.local_iter)
# Sort results and build the global return object
self.sort_result()
# Lowest values used to report in case of failures
self.f_lowest = self.res.fun
self.x_lowest = self.res.x
else:
self.find_lowest_vertex()
def find_lowest_vertex(self):
# Find the lowest objective function value on one of
# the vertices of the simplicial complex
if self.sampling_method == 'simplicial':
self.f_lowest = np.inf
for x in self.HC.V.cache:
if self.HC.V[x].f < self.f_lowest:
self.f_lowest = self.HC.V[x].f
self.x_lowest = self.HC.V[x].x_a
if self.f_lowest == np.inf: # no feasible point
self.f_lowest = None
self.x_lowest = None
else:
if self.fn == 0:
self.f_lowest = None
self.x_lowest = None
else:
self.f_I = np.argsort(self.F, axis=-1)
self.f_lowest = self.F[self.f_I[0]]
self.x_lowest = self.C[self.f_I[0]]
# Stopping criteria functions:
def finite_iterations(self):
if self.iters is not None:
if self.iters_done >= (self.iters - 1):
self.stop_global = True
if self.maxiter is not None: # Stop for infeasible sampling
if self.iters_done >= (self.maxiter - 1):
self.stop_global = True
return self.stop_global
def finite_fev(self):
# Finite function evals in the feasible domain
if self.fn >= self.maxfev:
self.stop_global = True
return self.stop_global
def finite_ev(self):
# Finite evaluations including infeasible sampling points
if self.n_sampled >= self.maxev:
self.stop_global = True
def finite_time(self):
if (time.time() - self.init) >= self.maxtime:
self.stop_global = True
def finite_precision(self):
"""
Stop the algorithm if the final function value is known
Specify in options (with ``self.f_min_true = options['f_min']``)
and the tolerance with ``f_tol = options['f_tol']``
"""
# If no minimizer has been found use the lowest sampling value
if len(self.LMC.xl_maps) == 0:
self.find_lowest_vertex()
# Function to stop algorithm at specified percentage error:
if self.f_lowest == 0.0:
if self.f_min_true == 0.0:
if self.f_lowest <= self.f_tol:
self.stop_global = True
else:
pe = (self.f_lowest - self.f_min_true) / abs(self.f_min_true)
if self.f_lowest <= self.f_min_true:
self.stop_global = True
# 2if (pe - self.f_tol) <= abs(1.0 / abs(self.f_min_true)):
if abs(pe) >= 2 * self.f_tol:
warnings.warn("A much lower value than expected f* =" +
" {} than".format(self.f_min_true) +
" the was found f_lowest =" +
"{} ".format(self.f_lowest))
if pe <= self.f_tol:
self.stop_global = True
return self.stop_global
def finite_homology_growth(self):
if self.LMC.size == 0:
return # pass on no reason to stop yet.
self.hgrd = self.LMC.size - self.hgr
self.hgr = self.LMC.size
if self.hgrd <= self.minhgrd:
self.stop_global = True
return self.stop_global
def stopping_criteria(self):
"""
Various stopping criteria ran every iteration
Returns
-------
stop : bool
"""
if self.maxiter is not None:
self.finite_iterations()
if self.iters is not None:
self.finite_iterations()
if self.maxfev is not None:
self.finite_fev()
if self.maxev is not None:
self.finite_ev()
if self.maxtime is not None:
self.finite_time()
if self.f_min_true is not None:
self.finite_precision()
if self.minhgrd is not None:
self.finite_homology_growth()
def iterate(self):
self.iterate_complex()
# Build minimizer pool
if self.minimize_every_iter:
if not self.break_routine:
self.find_minima() # Process minimizer pool
# Algorithm updates
self.iters_done += 1
def iterate_hypercube(self):
"""
Iterate a subdivision of the complex
Note: called with ``self.iterate_complex()`` after class initiation
"""
# Iterate the complex
if self.n_sampled == 0:
# Initial triangulation of the hyper-rectangle. Note that
# we use `self.raw_func` as `self.func` is a *wrapped* function
# that already takes the original function arguments into
# account.
self.HC = Complex(self.dim, self._raw_func, self.args,
self.symmetry, self.bounds, self.g_cons,
self.g_args)
else:
self.HC.split_generation()
# feasible sampling points counted by the triangulation.py routines
self.fn = self.HC.V.nfev
self.n_sampled = self.HC.V.size # nevs counted in triangulation.py
return
def iterate_delaunay(self):
"""
Build a complex of Delaunay triangulated points
Note: called with ``self.iterate_complex()`` after class initiation
"""
self.sampled_surface(infty_cons_sampl=self.infty_cons_sampl)
self.nc += self.n
self.n_sampled = self.nc
# Hypercube minimizers
def simplex_minimizers(self):
"""
Returns the indexes of all minimizers
"""
self.minimizer_pool = []
# Note: Can implement parallelization here
for x in self.HC.V.cache:
if self.HC.V[x].minimiser():
if self.disp:
logging.info('=' * 60)
logging.info(
'v.x = {} is minimizer'.format(self.HC.V[x].x_a))
logging.info('v.f = {} is minimizer'.format(self.HC.V[x].f))
logging.info('=' * 30)
if self.HC.V[x] not in self.minimizer_pool:
self.minimizer_pool.append(self.HC.V[x])
if self.disp:
logging.info('Neighbors:')
logging.info('=' * 30)
for vn in self.HC.V[x].nn:
logging.info('x = {} || f = {}'.format(vn.x, vn.f))
logging.info('=' * 60)
self.minimizer_pool_F = []
self.X_min = []
# normalized tuple in the Vertex cache
self.X_min_cache = {} # Cache used in hypercube sampling
for v in self.minimizer_pool:
self.X_min.append(v.x_a)
self.minimizer_pool_F.append(v.f)
self.X_min_cache[tuple(v.x_a)] = v.x
self.minimizer_pool_F = np.array(self.minimizer_pool_F)
self.X_min = np.array(self.X_min)
# TODO: Only do this if global mode
self.sort_min_pool()
return self.X_min
# Local minimization
# Minimizer pool processing
def minimise_pool(self, force_iter=False):
"""
This processing method can optionally minimise only the best candidate
solutions in the minimizer pool
Parameters
----------
force_iter : int
Number of starting minimizers to process (can be sepcified
globally or locally)
"""
# Find first local minimum
# NOTE: Since we always minimize this value regardless it is a waste to
# build the topograph first before minimizing
lres_f_min = self.minimize(self.X_min[0], ind=self.minimizer_pool[0])
# Trim minimized point from current minimizer set
self.trim_min_pool(0)
# Force processing to only
if force_iter:
self.local_iter = force_iter
while not self.stop_l_iter:
# Global stopping criteria:
if self.f_min_true is not None:
if (lres_f_min.fun - self.f_min_true) / abs(
self.f_min_true) <= self.f_tol:
self.stop_l_iter = True
break
# Note first iteration is outside loop:
if self.local_iter is not None:
if self.disp:
logging.info(
'SHGO.iters in function minimise_pool = {}'.format(
self.local_iter))
self.local_iter -= 1
if self.local_iter == 0:
self.stop_l_iter = True
break
if np.shape(self.X_min)[0] == 0:
self.stop_l_iter = True
break
# Construct topograph from current minimizer set
# (NOTE: This is a very small topograph using only the minizer pool
# , it might be worth using some graph theory tools instead.
self.g_topograph(lres_f_min.x, self.X_min)
# Find local minimum at the miniser with the greatest Euclidean
# distance from the current solution
ind_xmin_l = self.Z[:, -1]
lres_f_min = self.minimize(self.Ss[-1, :], self.minimizer_pool[-1])
# Trim minimised point from current minimizer set
self.trim_min_pool(ind_xmin_l)
# Reset controls
self.stop_l_iter = False
return
def sort_min_pool(self):
# Sort to find minimum func value in min_pool
self.ind_f_min = np.argsort(self.minimizer_pool_F)
self.minimizer_pool = np.array(self.minimizer_pool)[self.ind_f_min]
self.minimizer_pool_F = np.array(self.minimizer_pool_F)[
self.ind_f_min]
return
def trim_min_pool(self, trim_ind):
self.X_min = np.delete(self.X_min, trim_ind, axis=0)
self.minimizer_pool_F = np.delete(self.minimizer_pool_F, trim_ind)
self.minimizer_pool = np.delete(self.minimizer_pool, trim_ind)
return
def g_topograph(self, x_min, X_min):
"""
Returns the topographical vector stemming from the specified value
``x_min`` for the current feasible set ``X_min`` with True boolean
values indicating positive entries and False values indicating
negative entries.
"""
x_min = np.array([x_min])
self.Y = spatial.distance.cdist(x_min, X_min, 'euclidean')
# Find sorted indexes of spatial distances:
self.Z = np.argsort(self.Y, axis=-1)
self.Ss = X_min[self.Z][0]
self.minimizer_pool = self.minimizer_pool[self.Z]
self.minimizer_pool = self.minimizer_pool[0]
return self.Ss
# Local bound functions
def construct_lcb_simplicial(self, v_min):
"""
Construct locally (approximately) convex bounds
Parameters
----------
v_min : Vertex object
The minimizer vertex
Returns
-------
cbounds : list of lists
List of size dimension with length-2 list of bounds for each dimension
"""
cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds]
# Loop over all bounds
for vn in v_min.nn:
for i, x_i in enumerate(vn.x_a):
# Lower bound
if (x_i < v_min.x_a[i]) and (x_i > cbounds[i][0]):
cbounds[i][0] = x_i
# Upper bound
if (x_i > v_min.x_a[i]) and (x_i < cbounds[i][1]):
cbounds[i][1] = x_i
if self.disp:
logging.info('cbounds found for v_min.x_a = {}'.format(v_min.x_a))
logging.info('cbounds = {}'.format(cbounds))
return cbounds
def construct_lcb_delaunay(self, v_min, ind=None):
"""
Construct locally (approximately) convex bounds
Parameters
----------
v_min : Vertex object
The minimizer vertex
Returns
-------
cbounds : list of lists
List of size dimension with length-2 list of bounds for each dimension
"""
cbounds = [[x_b_i[0], x_b_i[1]] for x_b_i in self.bounds]
return cbounds
# Minimize a starting point locally
def minimize(self, x_min, ind=None):
"""
This function is used to calculate the local minima using the specified
sampling point as a starting value.
Parameters
----------
x_min : vector of floats
Current starting point to minimize.
Returns
-------
lres : OptimizeResult
The local optimization result represented as a `OptimizeResult`
object.
"""
# Use minima maps if vertex was already run
if self.disp:
logging.info('Vertex minimiser maps = {}'.format(self.LMC.v_maps))
if self.LMC[x_min].lres is not None:
return self.LMC[x_min].lres
# TODO: Check discarded bound rules
if self.callback is not None:
print('Callback for '
'minimizer starting at {}:'.format(x_min))
if self.disp:
print('Starting '
'minimization at {}...'.format(x_min))
if self.sampling_method == 'simplicial':
x_min_t = tuple(x_min)
# Find the normalized tuple in the Vertex cache:
x_min_t_norm = self.X_min_cache[tuple(x_min_t)]
x_min_t_norm = tuple(x_min_t_norm)
g_bounds = self.construct_lcb_simplicial(self.HC.V[x_min_t_norm])
if 'bounds' in self.min_solver_args:
self.minimizer_kwargs['bounds'] = g_bounds
else:
g_bounds = self.construct_lcb_delaunay(x_min, ind=ind)
if 'bounds' in self.min_solver_args:
self.minimizer_kwargs['bounds'] = g_bounds
if self.disp and 'bounds' in self.minimizer_kwargs:
print('bounds in kwarg:')
print(self.minimizer_kwargs['bounds'])
# Local minimization using scipy.optimize.minimize:
lres = minimize(self.func, x_min, **self.minimizer_kwargs)
if self.disp:
print('lres = {}'.format(lres))
# Local function evals for all minimizers
self.res.nlfev += lres.nfev
if 'njev' in lres:
self.res.nljev += lres.njev
if 'nhev' in lres:
self.res.nlhev += lres.nhev
try: # Needed because of the brain dead 1x1 NumPy arrays
lres.fun = lres.fun[0]
except (IndexError, TypeError):
lres.fun
# Append minima maps
self.LMC[x_min]
self.LMC.add_res(x_min, lres, bounds=g_bounds)
return lres
# Post local minimization processing
def sort_result(self):
"""
Sort results and build the global return object
"""
# Sort results in local minima cache
results = self.LMC.sort_cache_result()
self.res.xl = results['xl']
self.res.funl = results['funl']
self.res.x = results['x']
self.res.fun = results['fun']
# Add local func evals to sampling func evals
# Count the number of feasible vertices and add to local func evals:
self.res.nfev = self.fn + self.res.nlfev
return self.res
# Algorithm controls
def fail_routine(self, mes=("Failed to converge")):
self.break_routine = True
self.res.success = False
self.X_min = [None]
self.res.message = mes
def sampled_surface(self, infty_cons_sampl=False):
"""
Sample the function surface.
There are 2 modes, if ``infty_cons_sampl`` is True then the sampled
points that are generated outside the feasible domain will be
assigned an ``inf`` value in accordance with SHGO rules.
This guarantees convergence and usually requires less objective function
evaluations at the computational costs of more Delaunay triangulation
points.
If ``infty_cons_sampl`` is False, then the infeasible points are discarded
and only a subspace of the sampled points are used. This comes at the
cost of the loss of guaranteed convergence and usually requires more
objective function evaluations.
"""
# Generate sampling points
if self.disp:
print('Generating sampling points')
self.sampling(self.nc, self.dim)
self.n = self.nc
if not infty_cons_sampl:
# Find subspace of feasible points
if self.g_cons is not None:
self.sampling_subspace()
# Sort remaining samples
self.sorted_samples()
# Find objective function references
self.fun_ref()
self.n_sampled = self.nc
def delaunay_complex_minimisers(self):
# Construct complex minimizers on the current sampling set.
# if self.fn >= (self.dim + 1):
if self.fn >= (self.dim + 2):
# TODO: Check on strange Qhull error where the number of vertices
# required for an initial simplex is higher than n + 1?
if self.dim < 2: # Scalar objective functions
if self.disp:
print('Constructing 1-D minimizer pool')
self.ax_subspace()
self.surface_topo_ref()
self.minimizers_1D()
else: # Multivariate functions.
if self.disp:
print('Constructing Gabrial graph and minimizer pool')
if self.iters == 1:
self.delaunay_triangulation(grow=False)
else:
self.delaunay_triangulation(grow=True, n_prc=self.n_prc)
self.n_prc = self.C.shape[0]
if self.disp:
print('Triangulation completed, building minimizer pool')
self.delaunay_minimizers()
if self.disp:
logging.info(
"Minimizer pool = SHGO.X_min = {}".format(self.X_min))
else:
if self.disp:
print(
'Not enough sampling points found in the feasible domain.')
self.minimizer_pool = [None]
try:
self.X_min
except AttributeError:
self.X_min = []
def sampling_custom(self, n, dim):
"""
Generates uniform sampling points in a hypercube and scales the points
to the bound limits.
"""
# Generate sampling points.
# Generate uniform sample points in [0, 1]^m \subset R^m
self.C = self.sampling_function(n, dim)
# Distribute over bounds
for i in range(len(self.bounds)):
self.C[:, i] = (self.C[:, i] *
(self.bounds[i][1] - self.bounds[i][0])
+ self.bounds[i][0])
return self.C
def sampling_subspace(self):
"""Find subspace of feasible points from g_func definition"""
# Subspace of feasible points.
for ind, g in enumerate(self.g_cons):
self.C = self.C[g(self.C.T, *self.g_args[ind]) >= 0.0]
if self.C.size == 0:
self.res.message = ('No sampling point found within the '
+ 'feasible set. Increasing sampling '
+ 'size.')
# sampling correctly for both 1-D and >1-D cases
if self.disp:
print(self.res.message)
def sorted_samples(self): # Validated
"""Find indexes of the sorted sampling points"""
self.Ind_sorted = np.argsort(self.C, axis=0)
self.Xs = self.C[self.Ind_sorted]
return self.Ind_sorted, self.Xs
def ax_subspace(self): # Validated
"""
Finds the subspace vectors along each component axis.
"""
self.Ci = []
self.Xs_i = []
self.Ii = []
for i in range(self.dim):
self.Ci.append(self.C[:, i])
self.Ii.append(self.Ind_sorted[:, i])
self.Xs_i.append(self.Xs[:, i])
def fun_ref(self):
"""
Find the objective function output reference table
"""
# TODO: Replace with cached wrapper
# Note: This process can be pooled easily
# Obj. function returns to be used as reference table.:
f_cache_bool = False
if self.fn > 0: # Store old function evaluations
Ftemp = self.F
fn_old = self.fn
f_cache_bool = True
self.F = np.zeros(np.shape(self.C)[0])
# NOTE: It might be easier to replace this with a cached
# objective function
for i in range(self.fn, np.shape(self.C)[0]):
eval_f = True
if self.g_cons is not None:
for g in self.g_cons:
if g(self.C[i, :], *self.args) < 0.0:
eval_f = False
break # Breaks the g loop
if eval_f:
self.F[i] = self.func(self.C[i, :])
self.fn += 1
elif self.infty_cons_sampl:
self.F[i] = np.inf
self.fn += 1
if f_cache_bool:
if fn_old > 0: # Restore saved function evaluations
self.F[0:fn_old] = Ftemp
return self.F
def surface_topo_ref(self): # Validated
"""
Find the BD and FD finite differences along each component vector.
"""
# Replace numpy inf, -inf and nan objects with floating point numbers
# nan --> float
self.F[np.isnan(self.F)] = np.inf
# inf, -inf --> floats
self.F = np.nan_to_num(self.F)
self.Ft = self.F[self.Ind_sorted]
self.Ftp = np.diff(self.Ft, axis=0) # FD
self.Ftm = np.diff(self.Ft[::-1], axis=0)[::-1] # BD
def sample_topo(self, ind):
# Find the position of the sample in the component axial directions
self.Xi_ind_pos = []
self.Xi_ind_topo_i = []
for i in range(self.dim):
for x, I_ind in zip(self.Ii[i], range(len(self.Ii[i]))):
if x == ind:
self.Xi_ind_pos.append(I_ind)
# Use the topo reference tables to find if point is a minimizer on
# the current axis
# First check if index is on the boundary of the sampling points:
if self.Xi_ind_pos[i] == 0:
# if boundary is in basin
self.Xi_ind_topo_i.append(self.Ftp[:, i][0] > 0)
elif self.Xi_ind_pos[i] == self.fn - 1:
# Largest value at sample size
self.Xi_ind_topo_i.append(self.Ftp[:, i][self.fn - 2] < 0)
# Find axial reference for other points
else:
Xi_ind_top_p = self.Ftp[:, i][self.Xi_ind_pos[i]] > 0
Xi_ind_top_m = self.Ftm[:, i][self.Xi_ind_pos[i] - 1] > 0
self.Xi_ind_topo_i.append(Xi_ind_top_p and Xi_ind_top_m)
if np.array(self.Xi_ind_topo_i).all():
self.Xi_ind_topo = True
else:
self.Xi_ind_topo = False
self.Xi_ind_topo = np.array(self.Xi_ind_topo_i).all()
return self.Xi_ind_topo
def minimizers_1D(self):
"""
Returns the indices of all minimizers
"""
self.minimizer_pool = []
# Note: Can implement parallelization here
for ind in range(self.fn):
min_bool = self.sample_topo(ind)
if min_bool:
self.minimizer_pool.append(ind)
self.minimizer_pool_F = self.F[self.minimizer_pool]
# Sort to find minimum func value in min_pool
self.sort_min_pool()
if not len(self.minimizer_pool) == 0:
self.X_min = self.C[self.minimizer_pool]
# If function is called again and pool is found unbreak:
else:
self.X_min = []
return self.X_min
def delaunay_triangulation(self, grow=False, n_prc=0):
if not grow:
self.Tri = spatial.Delaunay(self.C)
else:
if hasattr(self, 'Tri'):
self.Tri.add_points(self.C[n_prc:, :])
else:
self.Tri = spatial.Delaunay(self.C, incremental=True)
return self.Tri
@staticmethod
def find_neighbors_delaunay(pindex, triang):
"""
Returns the indices of points connected to ``pindex`` on the Gabriel
chain subgraph of the Delaunay triangulation.
"""
return triang.vertex_neighbor_vertices[1][
triang.vertex_neighbor_vertices[0][pindex]:
triang.vertex_neighbor_vertices[0][pindex + 1]]
def sample_delaunay_topo(self, ind):
self.Xi_ind_topo_i = []
# Find the position of the sample in the component Gabrial chain
G_ind = self.find_neighbors_delaunay(ind, self.Tri)
# Find finite deference between each point
for g_i in G_ind:
rel_topo_bool = self.F[ind] < self.F[g_i]
self.Xi_ind_topo_i.append(rel_topo_bool)
# Check if minimizer
self.Xi_ind_topo = np.array(self.Xi_ind_topo_i).all()
return self.Xi_ind_topo
def delaunay_minimizers(self):
"""
Returns the indices of all minimizers
"""
self.minimizer_pool = []
# Note: Can easily be parralized
if self.disp:
logging.info('self.fn = {}'.format(self.fn))
logging.info('self.nc = {}'.format(self.nc))
logging.info('np.shape(self.C)'
' = {}'.format(np.shape(self.C)))
for ind in range(self.fn):
min_bool = self.sample_delaunay_topo(ind)
if min_bool:
self.minimizer_pool.append(ind)
self.minimizer_pool_F = self.F[self.minimizer_pool]
# Sort to find minimum func value in min_pool
self.sort_min_pool()
if self.disp:
logging.info('self.minimizer_pool = {}'.format(self.minimizer_pool))
if not len(self.minimizer_pool) == 0:
self.X_min = self.C[self.minimizer_pool]
else:
self.X_min = [] # Empty pool breaks main routine
return self.X_min
class LMap:
def __init__(self, v):
self.v = v
self.x_l = None
self.lres = None
self.f_min = None
self.lbounds = []
class LMapCache:
def __init__(self):
self.cache = {}
# Lists for search queries
self.v_maps = []
self.xl_maps = []
self.f_maps = []
self.lbound_maps = []
self.size = 0
def __getitem__(self, v):
v = np.ndarray.tolist(v)
v = tuple(v)
try:
return self.cache[v]
except KeyError:
xval = LMap(v)
self.cache[v] = xval
return self.cache[v]
def add_res(self, v, lres, bounds=None):
v = np.ndarray.tolist(v)
v = tuple(v)
self.cache[v].x_l = lres.x
self.cache[v].lres = lres
self.cache[v].f_min = lres.fun
self.cache[v].lbounds = bounds
# Update cache size
self.size += 1
# Cache lists for search queries
self.v_maps.append(v)
self.xl_maps.append(lres.x)
self.f_maps.append(lres.fun)
self.lbound_maps.append(bounds)
def sort_cache_result(self):
"""
Sort results and build the global return object
"""
results = {}
# Sort results and save
self.xl_maps = np.array(self.xl_maps)
self.f_maps = np.array(self.f_maps)
# Sorted indexes in Func_min
ind_sorted = np.argsort(self.f_maps)
# Save ordered list of minima
results['xl'] = self.xl_maps[ind_sorted] # Ordered x vals
self.f_maps = np.array(self.f_maps)
results['funl'] = self.f_maps[ind_sorted]
results['funl'] = results['funl'].T
# Find global of all minimizers
results['x'] = self.xl_maps[ind_sorted[0]] # Save global minima
results['fun'] = self.f_maps[ind_sorted[0]] # Save global fun value
self.xl_maps = np.ndarray.tolist(self.xl_maps)
self.f_maps = np.ndarray.tolist(self.f_maps)
return results
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