Fetching contributors…
Cannot retrieve contributors at this time
497 lines (420 sloc) 17.7 KB
This module implements the Sequential Least SQuares Programming optimization
algorithm (SLSQP), originally developed by Dieter Kraft.
.. autosummary::
:toctree: generated/
from __future__ import division, print_function, absolute_import
__all__ = ['approx_jacobian','fmin_slsqp']
import numpy as np
from scipy.optimize._slsqp import slsqp
from numpy import zeros, array, linalg, append, asfarray, concatenate, finfo, \
sqrt, vstack, exp, inf, where, isfinite, atleast_1d
from .optimize import wrap_function, OptimizeResult, _check_unknown_options
__docformat__ = "restructuredtext en"
_epsilon = sqrt(finfo(float).eps)
def approx_jacobian(x,func,epsilon,*args):
Approximate the Jacobian matrix of a callable function.
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.
An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
of the outputs of `func`, and ``lenx`` is the number of elements in
The approximation is done using forward differences.
x0 = asfarray(x)
f0 = atleast_1d(func(*((x0,)+args)))
jac = zeros([len(x0),len(f0)])
dx = zeros(len(x0))
for i in range(len(x0)):
dx[i] = epsilon
jac[i] = (func(*((x0+dx,)+args)) - f0)/epsilon
dx[i] = 0.0
return jac.transpose()
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,
Minimize a function using Sequential Least SQuares Programming
Python interface function for the SLSQP Optimization subroutine
originally implemented by Dieter Kraft.
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
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
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
Over-rides the iprint interface (preferred).
full_output : bool, optional
If False, return only the minimizer of func (default).
Otherwise, output final objective function and summary
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.
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.
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 exceeded
Examples are given :ref:`in the tutorial <tutorial-sqlsp>`.
if disp is not None:
iprint = disp
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']
return res['x']
def _minimize_slsqp(func, x0, args=(), jac=None, bounds=None,
maxiter=100, ftol=1.0E-6, iprint=1, disp=False,
eps=_epsilon, callback=None,
Minimize a scalar function of one or more variables using Sequential
Least SQuares Programming (SLSQP).
ftol : float
Precision goal for the value of f in the stopping criterion.
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
Maximum number of iterations.
fprime = jac
iter = maxiter
acc = ftol
epsilon = eps
if not disp:
iprint = 0
# Constraints are triaged per type into a dictionnary of tuples
if isinstance(constraints, dict):
constraints = (constraints, )
cons = {'eq': (), 'ineq': ()}
for ic, con in enumerate(constraints):
# check type
ctype = con['type'].lower()
except KeyError:
raise KeyError('Constraint %d has no type defined.' % ic)
except TypeError:
raise TypeError('Constraints must be defined using a '
except AttributeError:
raise TypeError("Constraint's type must be a string.")
if ctype not in ['eq', 'ineq']:
raise ValueError("Unknown constraint type '%s'." % con['type'])
# check function
if 'fun' not in con:
raise ValueError('Constraint %d has no function defined.' % ic)
# 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):
return approx_jacobian(x, fun, epsilon, *args)
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 exceeded"}
# Wrap func
feval, func = wrap_function(func, args)
# Wrap fprime, if provided, or approx_jacobian if not
if fprime:
geval, fprime = wrap_function(fprime, args)
geval, fprime = wrap_function(approx_jacobian, (func, epsilon))
# Transform x0 into an array.
x = asfarray(x0).flatten()
# Set the parameters that SLSQP will need
# meq, mieq: number of equality and inequality constraints
meq = sum(map(len, [atleast_1d(c['fun'](x, *c['args'])) for c in cons['eq']]))
mieq = sum(map(len, [atleast_1d(c['fun'](x, *c['args'])) for c in cons['ineq']]))
# m = The total number of constraints
m = meq + mieq
# la = The number of constraints, or 1 if there are no constraints
la = array([1, m]).max()
# n = The number of independent variables
n = len(x)
# Define the workspaces for SLSQP
n1 = n + 1
mineq = m - meq + n1 + n1
len_w = (3*n1+m)*(n1+1)+(n1-meq+1)*(mineq+2) + 2*mineq+(n1+mineq)*(n1-meq) \
+ 2*meq + n1 + ((n+1)*n)//2 + 2*m + 3*n + 3*n1 + 1
len_jw = mineq
w = zeros(len_w)
jw = zeros(len_jw)
# 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)
bnds = array(bounds, float)
if bnds.shape[0] != n:
raise IndexError('SLSQP Error: the length of bounds is not '
'compatible with that of x0.')
bnderr = where(bnds[:, 0] > bnds[:, 1])[0]
if bnderr.any():
raise ValueError('SLSQP Error: lb > ub in bounds %s.' %
', '.join(str(b) for b in bnderr))
xl, xu = bnds[:, 0], bnds[:, 1]
# Mark infinite bounds with nans; the Fortran code understands this
infbnd = ~isfinite(bnds)
xl[infbnd[:, 0]] = np.nan
xu[infbnd[:, 1]] = np.nan
# Initialize the iteration counter and the mode value
mode = array(0,int)
acc = array(acc,float)
majiter = array(iter,int)
majiter_prev = 0
# Print the header if iprint >= 2
if iprint >= 2:
print("%5s %5s %16s %16s" % ("NIT","FC","OBJFUN","GNORM"))
while 1:
if mode == 0 or mode == 1: # objective and constraint evaluation requird
# Compute objective function
fx = float(np.asarray(func(x)))
raise ValueError("Objective function must return a scalar")
# Compute the constraints
if cons['eq']:
c_eq = concatenate([atleast_1d(con['fun'](x, *con['args']))
for con in cons['eq']])
c_eq = zeros(0)
if cons['ineq']:
c_ieq = concatenate([atleast_1d(con['fun'](x, *con['args']))
for con in cons['ineq']])
c_ieq = zeros(0)
# Now combine c_eq and c_ieq into a single matrix
c = concatenate((c_eq, c_ieq))
if mode == 0 or mode == -1: # gradient evaluation required
# Compute the derivatives of the objective function
# For some reason SLSQP wants g dimensioned to n+1
g = append(fprime(x),0.0)
# Compute the normals of the constraints
if cons['eq']:
a_eq = vstack([con['jac'](x, *con['args'])
for con in cons['eq']])
else: # no equality constraint
a_eq = zeros((meq, n))
if cons['ineq']:
a_ieq = vstack([con['jac'](x, *con['args'])
for con in cons['ineq']])
else: # no inequality constraint
a_ieq = zeros((mieq, n))
# Now combine a_eq and a_ieq into a single a matrix
if m == 0: # no constraints
a = zeros((la, n))
a = vstack((a_eq, a_ieq))
a = concatenate((a,zeros([la,1])),1)
# Call SLSQP
slsqp(m, meq, x, xl, xu, fx, c, g, a, acc, majiter, mode, w, jw)
# call callback if major iteration has incremented
if callback is not None and majiter > majiter_prev:
# Print the status of the current iterate if iprint > 2 and the
# major iteration has incremented
if iprint >= 2 and majiter > majiter_prev:
print("%5i %5i % 16.6E % 16.6E" % (majiter,feval[0],
# If exit mode is not -1 or 1, slsqp has completed
if abs(mode) != 1:
majiter_prev = int(majiter)
# Optimization loop complete. Print status if requested
if iprint >= 1:
print(exit_modes[int(mode)] + " (Exit mode " + str(mode) + ')')
print(" Current function value:", fx)
print(" Iterations:", majiter)
print(" Function evaluations:", feval[0])
print(" Gradient evaluations:", geval[0])
return OptimizeResult(x=x, fun=fx, jac=g[:-1], nit=int(majiter),
nfev=feval[0], njev=geval[0], status=int(mode),
message=exit_modes[int(mode)], success=(mode == 0))
if __name__ == '__main__':
# objective function
def fun(x, r=[4, 2, 4, 2, 1]):
""" Objective function """
return exp(x[0]) * (r[0] * x[0]**2 + r[1] * x[1]**2 +
r[2] * x[0] * x[1] + r[3] * x[1] +
# bounds
bnds = array([[-inf]*2, [inf]*2]).T
bnds[:, 0] = [0.1, 0.2]
# constraints
def feqcon(x, b=1):
""" Equality constraint """
return array([x[0]**2 + x[1] - b])
def jeqcon(x, b=1):
""" Jacobian of equality constraint """
return array([[2*x[0], 1]])
def fieqcon(x, c=10):
""" Inequality constraint """
return array([x[0] * x[1] + c])
def jieqcon(x, c=10):
""" Jacobian of Inequality constraint """
return array([[1, 1]])
# constraints dictionaries
cons = ({'type': 'eq', 'fun': feqcon, 'jac': jeqcon, 'args': (1, )},
{'type': 'ineq', 'fun': fieqcon, 'jac': jieqcon, 'args': (10,)})
# Bounds constraint problem
print(' Bounds constraints '.center(72, '-'))
print(' * fmin_slsqp')
x, f = fmin_slsqp(fun, array([-1, 1]), bounds=bnds, disp=1,
print(' * _minimize_slsqp')
res = _minimize_slsqp(fun, array([-1, 1]), bounds=bnds,
**{'disp': True})
# Equality and inequality constraints problem
print(' Equality and inequality constraints '.center(72, '-'))
print(' * fmin_slsqp')
x, f = fmin_slsqp(fun, array([-1, 1]),
f_eqcons=feqcon, fprime_eqcons=jeqcon,
f_ieqcons=fieqcon, fprime_ieqcons=jieqcon,
disp=1, full_output=True)[:2]
print(' * _minimize_slsqp')
res = _minimize_slsqp(fun, array([-1, 1]), constraints=cons,
**{'disp': True})