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webapp/seed/Lib/site-packages/scipy/optimize/_slsqp_py.py

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"""
This module implements the Sequential Least Squares Programming optimization
algorithm (SLSQP), originally developed by Dieter Kraft.
See http://www.netlib.org/toms/733
Functions
---------
.. autosummary::
:toctree: generated/
approx_jacobian
fmin_slsqp
"""
__all__ = ['approx_jacobian', 'fmin_slsqp']
import numpy as np
from ._slsqplib import slsqp
from scipy.linalg import norm as lanorm
from ._optimize import (OptimizeResult, _check_unknown_options,
_prepare_scalar_function, _clip_x_for_func,
_check_clip_x, _wrap_callback)
from ._numdiff import approx_derivative
from ._constraints import old_bound_to_new, _arr_to_scalar
from scipy._lib._array_api import array_namespace
from scipy._lib import array_api_extra as xpx
from scipy._lib._util import _call_callback_maybe_halt
from numpy.typing import NDArray
__docformat__ = "restructuredtext en"
_epsilon = np.sqrt(np.finfo(np.float64).eps)
def approx_jacobian(x, func, epsilon, *args):
"""
Approximate the Jacobian matrix of a callable function.
Parameters
----------
x : array_like
The state vector at which to compute the Jacobian matrix.
func : callable f(x,*args)
The vector-valued function.
epsilon : float
The perturbation used to determine the partial derivatives.
args : sequence
Additional arguments passed to func.
Returns
-------
An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
of the outputs of `func`, and ``lenx`` is the number of elements in
`x`.
Notes
-----
The approximation is done using forward differences.
"""
# approx_derivative returns (m, n) == (lenf, lenx)
jac = approx_derivative(func, x, method='2-point', abs_step=epsilon,
args=args)
# if func returns a scalar jac.shape will be (lenx,). Make sure
# it's at least a 2D array.
return np.atleast_2d(jac)
def fmin_slsqp(func, x0, eqcons=(), f_eqcons=None, ieqcons=(), f_ieqcons=None,
bounds=(), fprime=None, fprime_eqcons=None,
fprime_ieqcons=None, args=(), iter=100, acc=1.0E-6,
iprint=1, disp=None, full_output=0, epsilon=_epsilon,
callback=None):
"""
Minimize a function using Sequential Least Squares Programming
Python interface function for the SLSQP Optimization subroutine
originally implemented by Dieter Kraft.
Parameters
----------
func : callable f(x,*args)
Objective function. Must return a scalar.
x0 : 1-D ndarray of float
Initial guess for the independent variable(s).
eqcons : list, optional
A list of functions of length n such that
eqcons[j](x,*args) == 0.0 in a successfully optimized
problem.
f_eqcons : callable f(x,*args), optional
Returns a 1-D array in which each element must equal 0.0 in a
successfully optimized problem. If f_eqcons is specified,
eqcons is ignored.
ieqcons : list, optional
A list of functions of length n such that
ieqcons[j](x,*args) >= 0.0 in a successfully optimized
problem.
f_ieqcons : callable f(x,*args), optional
Returns a 1-D ndarray in which each element must be greater or
equal to 0.0 in a successfully optimized problem. If
f_ieqcons is specified, ieqcons is ignored.
bounds : list, optional
A list of tuples specifying the lower and upper bound
for each independent variable [(xl0, xu0),(xl1, xu1),...]
Infinite values will be interpreted as large floating values.
fprime : callable ``f(x,*args)``, optional
A function that evaluates the partial derivatives of func.
fprime_eqcons : callable ``f(x,*args)``, optional
A function of the form ``f(x, *args)`` that returns the m by n
array of equality constraint normals. If not provided,
the normals will be approximated. The array returned by
fprime_eqcons should be sized as ( len(eqcons), len(x0) ).
fprime_ieqcons : callable ``f(x,*args)``, optional
A function of the form ``f(x, *args)`` that returns the m by n
array of inequality constraint normals. If not provided,
the normals will be approximated. The array returned by
fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ).
args : sequence, optional
Additional arguments passed to func and fprime.
iter : int, optional
The maximum number of iterations.
acc : float, optional
Requested accuracy.
iprint : int, optional
The verbosity of fmin_slsqp :
* iprint <= 0 : Silent operation
* iprint == 1 : Print summary upon completion (default)
* iprint >= 2 : Print status of each iterate and summary
disp : int, optional
Overrides the iprint interface (preferred).
full_output : bool, optional
If False, return only the minimizer of func (default).
Otherwise, output final objective function and summary
information.
epsilon : float, optional
The step size for finite-difference derivative estimates.
callback : callable, optional
Called after each iteration, as ``callback(x)``, where ``x`` is the
current parameter vector.
Returns
-------
out : ndarray of float
The final minimizer of func.
fx : ndarray of float, if full_output is true
The final value of the objective function.
its : int, if full_output is true
The number of iterations.
imode : int, if full_output is true
The exit mode from the optimizer (see below).
smode : string, if full_output is true
Message describing the exit mode from the optimizer.
See also
--------
minimize: Interface to minimization algorithms for multivariate
functions. See the 'SLSQP' `method` in particular.
Notes
-----
Exit modes are defined as follows:
- ``-1`` : Gradient evaluation required (g & a)
- ``0`` : Optimization terminated successfully
- ``1`` : Function evaluation required (f & c)
- ``2`` : More equality constraints than independent variables
- ``3`` : More than 3*n iterations in LSQ subproblem
- ``4`` : Inequality constraints incompatible
- ``5`` : Singular matrix E in LSQ subproblem
- ``6`` : Singular matrix C in LSQ subproblem
- ``7`` : Rank-deficient equality constraint subproblem HFTI
- ``8`` : Positive directional derivative for linesearch
- ``9`` : Iteration limit reached
Examples
--------
Examples are given :ref:`in the tutorial <tutorial-sqlsp>`.
"""
if disp is not None:
iprint = disp
# selects whether to use callback(x) or callback(intermediate_result)
callback = _wrap_callback(callback, "slsqp")
opts = {'maxiter': iter,
'ftol': acc,
'iprint': iprint,
'disp': iprint != 0,
'eps': epsilon,
'callback': callback}
# Build the constraints as a tuple of dictionaries
cons = ()
# 1. constraints of the 1st kind (eqcons, ieqcons); no Jacobian; take
# the same extra arguments as the objective function.
cons += tuple({'type': 'eq', 'fun': c, 'args': args} for c in eqcons)
cons += tuple({'type': 'ineq', 'fun': c, 'args': args} for c in ieqcons)
# 2. constraints of the 2nd kind (f_eqcons, f_ieqcons) and their Jacobian
# (fprime_eqcons, fprime_ieqcons); also take the same extra arguments
# as the objective function.
if f_eqcons:
cons += ({'type': 'eq', 'fun': f_eqcons, 'jac': fprime_eqcons,
'args': args}, )
if f_ieqcons:
cons += ({'type': 'ineq', 'fun': f_ieqcons, 'jac': fprime_ieqcons,
'args': args}, )
res = _minimize_slsqp(func, x0, args, jac=fprime, bounds=bounds,
constraints=cons, **opts)
if full_output:
return res['x'], res['fun'], res['nit'], res['status'], res['message']
else:
return res['x']
def _minimize_slsqp(func, x0, args=(), jac=None, bounds=None,
constraints=(),
maxiter=100, ftol=1.0E-6, iprint=1, disp=False,
eps=_epsilon, callback=None, finite_diff_rel_step=None,
workers=None, **unknown_options):
"""
Minimize a scalar function of one or more variables using Sequential
Least Squares Programming (SLSQP).
Parameters
----------
ftol : float
Precision target for the value of f in the stopping criterion. This value
controls the final accuracy for checking various optimality conditions;
gradient of the lagrangian and absolute sum of the constraint violations
should be lower than ``ftol``. Similarly, computed step size and the
objective function changes are checked against this value. Default is 1e-6.
eps : float
Step size used for numerical approximation of the Jacobian.
disp : bool
Set to True to print convergence messages. If False,
`verbosity` is ignored and set to 0.
maxiter : int, optional
Maximum number of iterations. Default value is 100.
finite_diff_rel_step : None or array_like, optional
If ``jac in ['2-point', '3-point', 'cs']`` the relative step size to
use for numerical approximation of `jac`. The absolute step
size is computed as ``h = rel_step * sign(x) * max(1, abs(x))``,
possibly adjusted to fit into the bounds. For ``method='3-point'``
the sign of `h` is ignored. If None (default) then step is selected
automatically.
workers : int, map-like callable, optional
A map-like callable, such as `multiprocessing.Pool.map` for evaluating
any numerical differentiation in parallel.
This evaluation is carried out as ``workers(fun, iterable)``.
.. versionadded:: 1.16.0
Returns
-------
res : OptimizeResult
The optimization result represented as an `OptimizeResult` object.
In this dict-like object the following fields are of particular importance:
``x`` the solution array, ``success`` a Boolean flag indicating if the
optimizer exited successfully, ``message`` which describes the reason for
termination, and ``multipliers`` which contains the Karush-Kuhn-Tucker
(KKT) multipliers for the QP approximation used in solving the original
nonlinear problem. See ``Notes`` below. See also `OptimizeResult` for a
description of other attributes.
Notes
-----
The KKT multipliers are returned in the ``OptimizeResult.multipliers``
attribute as a NumPy array. Denoting the dimension of the equality constraints
with ``meq``, and of inequality constraints with ``mineq``, then the returned
array slice ``m[:meq]`` contains the multipliers for the equality constraints,
and the remaining ``m[meq:meq + mineq]`` contains the multipliers for the
inequality constraints. The multipliers corresponding to bound inequalities
are not returned. See [1]_ pp. 321 or [2]_ for an explanation of how to interpret
these multipliers. The internal QP problem is solved using the methods given
in [3]_ Chapter 25.
Note that if new-style `NonlinearConstraint` or `LinearConstraint` were
used, then ``minimize`` converts them first to old-style constraint dicts.
It is possible for a single new-style constraint to simultaneously contain
both inequality and equality constraints. This means that if there is mixing
within a single constraint, then the returned list of multipliers will have
a different length than the original new-style constraints.
References
----------
.. [1] Nocedal, J., and S J Wright, 2006, "Numerical Optimization", Springer,
New York.
.. [2] Kraft, D., "A software package for sequential quadratic programming",
1988, Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center, Germany.
.. [3] Lawson, C. L., and R. J. Hanson, 1995, "Solving Least Squares Problems",
SIAM, Philadelphia, PA.
"""
_check_unknown_options(unknown_options)
acc = ftol
epsilon = eps
if not disp:
iprint = 0
# Transform x0 into an array.
xp = array_namespace(x0)
x0 = xpx.atleast_nd(xp.asarray(x0), ndim=1, xp=xp)
dtype = xp.float64
if xp.isdtype(x0.dtype, "real floating"):
dtype = x0.dtype
x = xp.reshape(xp.astype(x0, dtype), -1)
# SLSQP is sent 'old-style' bounds, 'new-style' bounds are required by
# ScalarFunction
if bounds is None or len(bounds) == 0:
new_bounds = (-np.inf, np.inf)
else:
new_bounds = old_bound_to_new(bounds)
# clip the initial guess to bounds, otherwise ScalarFunction doesn't work
x = np.clip(x, new_bounds[0], new_bounds[1])
# Constraints are triaged per type into a dictionary of tuples
if isinstance(constraints, dict):
constraints = (constraints, )
cons = {'eq': (), 'ineq': ()}
for ic, con in enumerate(constraints):
# check type
try:
ctype = con['type'].lower()
except KeyError as e:
raise KeyError(f'Constraint {ic} has no type defined.') from e
except TypeError as e:
raise TypeError('Constraints must be defined using a '
'dictionary.') from e
except AttributeError as e:
raise TypeError("Constraint's type must be a string.") from e
else:
if ctype not in ['eq', 'ineq']:
raise ValueError(f"Unknown constraint type '{con['type']}'.")
# check function
if 'fun' not in con:
raise ValueError(f'Constraint {ic} has no function defined.')
# check Jacobian
cjac = con.get('jac')
if cjac is None:
# approximate Jacobian function. The factory function is needed
# to keep a reference to `fun`, see gh-4240.
def cjac_factory(fun):
def cjac(x, *args):
x = _check_clip_x(x, new_bounds)
if jac in ['2-point', '3-point', 'cs']:
return approx_derivative(fun, x, method=jac, args=args,
rel_step=finite_diff_rel_step,
bounds=new_bounds)
else:
return approx_derivative(fun, x, method='2-point',
abs_step=epsilon, args=args,
bounds=new_bounds)
return cjac
cjac = cjac_factory(con['fun'])
# update constraints' dictionary
cons[ctype] += ({'fun': con['fun'],
'jac': cjac,
'args': con.get('args', ())}, )
exit_modes = {-1: "Gradient evaluation required (g & a)",
0: "Optimization terminated successfully",
1: "Function evaluation required (f & c)",
2: "More equality constraints than independent variables",
3: "More than 3*n iterations in LSQ subproblem",
4: "Inequality constraints incompatible",
5: "Singular matrix E in LSQ subproblem",
6: "Singular matrix C in LSQ subproblem",
7: "Rank-deficient equality constraint subproblem HFTI",
8: "Positive directional derivative for linesearch",
9: "Iteration limit reached"}
# Set the parameters that SLSQP will need
# meq, mieq: number of equality and inequality constraints
meq = sum(map(len, [np.atleast_1d(c['fun'](x, *c['args']))
for c in cons['eq']]))
mieq = sum(map(len, [np.atleast_1d(c['fun'](x, *c['args']))
for c in cons['ineq']]))
# m = The total number of constraints
m = meq + mieq
# n = The number of independent variables
n = len(x)
# Decompose bounds into xl and xu
if bounds is None or len(bounds) == 0:
xl = np.empty(n, dtype=float)
xu = np.empty(n, dtype=float)
xl.fill(np.nan)
xu.fill(np.nan)
else:
bnds = np.array([(_arr_to_scalar(lo), _arr_to_scalar(up))
for (lo, up) in bounds], float)
if bnds.shape[0] != n:
raise IndexError('SLSQP Error: the length of bounds is not '
'compatible with that of x0.')
with np.errstate(invalid='ignore'):
bnderr = bnds[:, 0] > bnds[:, 1]
if bnderr.any():
raise ValueError("SLSQP Error: lb > ub in bounds "
f"{', '.join(str(b) for b in bnderr)}.")
xl, xu = bnds[:, 0].copy(), bnds[:, 1].copy()
# Mark infinite bounds with nans; the C code expects this
infbnd = ~np.isfinite(bnds)
xl[infbnd[:, 0]] = np.nan
xu[infbnd[:, 1]] = np.nan
# ScalarFunction provides function and gradient evaluation
sf = _prepare_scalar_function(func, x, jac=jac, args=args, epsilon=eps,
finite_diff_rel_step=finite_diff_rel_step,
bounds=new_bounds, workers=workers)
# gh11403 SLSQP sometimes exceeds bounds by 1 or 2 ULP, make sure this
# doesn't get sent to the func/grad evaluator.
wrapped_fun = _clip_x_for_func(sf.fun, new_bounds)
wrapped_grad = _clip_x_for_func(sf.grad, new_bounds)
# Initialize internal SLSQP state variables dictionary
# This dictionary is passed to the SLSQP matching the C struct defined as
#
# struct SLSQP_static_vars {
# double acc, alpha, f0, gs, h1, h2, h3, h4, t, t0, tol;
# int exact, inconsistent, reset, iter, itermax, line, mode, meq;
# };
#
# exact : a dummy variable and should be kept 0 since the underlying code
# always uses an inexact search.
# inconsistent: a boolean set to 1 if the linearized QP is not well-defined
# while the original nonlinear problem is still solvable. Then
# the problem is augmented with a regularizing dummy variable.
# reset: holds the count of resetting bfgs to identity matrix.
# iter : the current and itermax is the maximum number of iterations.
# line : the current line search iteration.
# mode : the exit mode of the solver.
# alpha, f0, gs, h1, h2, h3, h4, t, t0 : internal variables used by the solver.
#
# The dict holds the intermediate state of the solver. The keys are the same
# as the C struct members and will be modified in-place.
state_dict = {
"acc": acc,
"alpha": 0.0,
"f0": 0.0,
"gs": 0.0,
"h1": 0.0,
"h2": 0.0,
"h3": 0.0,
"h4": 0.0,
"t": 0.0,
"t0": 0.0,
"tol": 10.0*acc,
"exact": 0,
"inconsistent": 0,
"reset": 0,
"iter": 0,
"itermax": int(maxiter),
"line": 0,
"m": m,
"meq": meq,
"mode": 0,
"n": n
}
# Print the header if iprint >= 2
if iprint >= 2:
print(f"{'NIT':>5} {'FC':>5} {'OBJFUN':>16} {'GNORM':>16}")
# Internal buffer and int array
indices = np.zeros([max(m + 2*n + 2, 1)], dtype=np.int32)
# The worst case workspace requirements for the buffer are:
# n*(n+1)//2 + m + 4*n + 3 # SLSQP
# (n+1)*(n+2) + (n+1)*meq + m + (mineq + 2*n + 2)*(n+1) + 3*n + 3 # LSQ
# mineq + 2n + 2 + 2*meq + (n+1) + (mineq + 3n + 3)*(n + 1 - meq) # LSEI
# (mineq + 2n + 2 + 2)*(n + 2) + mineq + 2n + 2 # LDP
# mineq + 2n + 2 # NNLS
# If we sum all up and simplify by the help of sympy we get the following
buffer_size = (
n*(n+1)//2 + 3*m*n - (m + 5*n + 7)*meq + 9*m + 8*n*n + 35*n + meq*meq + 28
)
# If no inequality constraints are given, top up workspace for the missing
# terms.
if mieq == 0:
buffer_size += 2*n*(n + 1)
buffer = np.zeros(max(buffer_size, 1), dtype=np.float64)
# mode is zero on entry, so call objective, constraints and gradients
# there should be no func evaluations here because it's cached from
# ScalarFunction
fx = wrapped_fun(x)
g = wrapped_grad(x)
# Allocate the multiplier array both for constraints and user specified
# bounds (extra +2 is for a possible augmented problem).
mult = np.zeros([max(1, m + 2*n + 2)], dtype=np.float64)
# Allocate the constraints and normals once and repopulate as needed
C = np.zeros([max(1, m), n], dtype=np.float64, order='F')
d = np.zeros([max(1, m)], dtype=np.float64)
_eval_con_normals(C, x, cons, m, meq)
_eval_constraint(d, x, cons, m, meq)
iter_prev = 0
while True:
# Call SLSQP
slsqp(state_dict, fx, g, C, d, x, mult, xl, xu, buffer, indices)
if state_dict['mode'] == 1: # objective and constraint evaluation required
fx = sf.fun(x)
_eval_constraint(d, x, cons, m, meq)
if state_dict['mode'] == -1: # gradient evaluation required
g = sf.grad(x)
_eval_con_normals(C, x, cons, m, meq)
if state_dict['iter'] > iter_prev:
# call callback if major iteration has incremented
if callback is not None:
intermediate_result = OptimizeResult(
x=np.copy(x),
fun=fx
)
if _call_callback_maybe_halt(callback, intermediate_result):
break
# Print the status of the current iterate if iprint > 2
if iprint >= 2:
print(f"{state_dict['iter']:5d} {sf.nfev:5d} "
f"{fx:16.6E} {lanorm(g):16.6E}")
# If exit mode is not -1 or 1, slsqp has completed
if abs(state_dict['mode']) != 1:
break
iter_prev = state_dict['iter']
# Optimization loop complete. Print status if requested
if iprint >= 1:
print(
exit_modes[state_dict['mode']] + f" (Exit mode {state_dict['mode']})"
)
print(" Current function value:", fx)
print(" Iterations:", state_dict['iter'])
print(" Function evaluations:", sf.nfev)
print(" Gradient evaluations:", sf.ngev)
return OptimizeResult(
x=x, fun=fx, jac=g, nit=state_dict['iter'], nfev=sf.nfev, njev=sf.ngev,
status=state_dict['mode'], message=exit_modes[state_dict['mode']],
success=(state_dict['mode'] == 0), multipliers=mult[:m]
)
# The following functions modify their first input argument in-place.
def _eval_constraint(d: NDArray, x: NDArray, cons: dict, m: int, meq: int):
if m == 0:
return
# The reason why we don't use regular increments with a sane for loop is that
# the constraint evaluations do not necessarily return scalars. Their
# output length needs to be taken into account while placing them in d.
if meq > 0:
row = 0
for con in cons['eq']:
temp = np.atleast_1d(con['fun'](x, *con['args'])).ravel()
d[row:row + len(temp)] = temp
row += len(temp)
if m > meq:
row = meq
for con in cons['ineq']:
temp = np.atleast_1d(con['fun'](x, *con['args'])).ravel()
d[row:row + len(temp)] = temp
row += len(temp)
return
def _eval_con_normals(C: NDArray, x: NDArray, cons: dict, m: int, meq: int):
if m == 0:
return
if meq > 0:
row = 0
for con in cons['eq']:
temp = np.atleast_2d(con['jac'](x, *con['args']))
C[row:row + temp.shape[0], :] = temp
row += temp.shape[0]
if m > meq:
row = meq
for con in cons['ineq']:
temp = np.atleast_2d(con['jac'](x, *con['args']))
C[row:row + temp.shape[0], :] = temp
row += temp.shape[0]
return
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