def minimize(fun, x0, args=(), method='Nelder-Mead', jac=None, hess=None,
hessp=None, options=dict(), full_output=False, callback=None,
retall=False):
"""
Minimization of scalar function of one or more variables.
Parameters
----------
fun : callable
Objective function.
x0 : ndarray
Initial guess.
args : tuple, optional
Extra arguments passed to the objective function and its
derivatives (Jacobian, Hessian).
method : str, optional
Type of solver. Should be one of:
{'Nelder-Mead', 'Powell', 'CG', 'BFGS', 'Newton-CG', 'Anneal'}.
jac : callable, optional
Jacobian of objective function (if None, Jacobian will be
estimated numerically). Only for CG, BFGS, Newton-CG.
hess, hessp : callable, optional
Hessian of objective function or Hessian of objective function
times an arbitrary vector p. Only for Newton-CG.
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. If it is not given, finite-differences
on `jac` are used to compute it.
options : dict, optional
A dictionary of solver options. All methods accept the following
generic options:
maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.
For method-specific options, see `show_minimize_options`.
full_output : bool, optional
If True, return optional outputs. Default is False.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
retall : bool, optional
If True, return a list of the solution at each iteration. This is only
done if `full_output` is True.
Returns
-------
xopt : ndarray
The solution.
info : dict
A dictionary of optional outputs (depending on the chosen method)
with the keys:
solution : ndarray
The solution (same as `xopt`).
success : bool
Boolean flag indicating if a solution was found.
status : int
An integer flag indicating the type of termination. Its
value depends on the underlying solver. Refer to `message`
for more information.
message : str
A string message giving information about the cause of the
termination.
fun, jac, hess : ndarray
Values of objective function, Jacobian and Hessian (if
available).
nfev, njev, nhev: int
Number of evaluations of the objective functions and of its
jacobian and hessian.
nit: int
Number of iterations.
direc: ndarray
Current set of direction vectors for the Powell method.
T : float
Final temperature for simulated annealing.
accept : int
Number of tests accepted.
allvecs : list
Solution at each iteration (if ``retall == True``).
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter.
Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This
algorithm has been successful in many applications but other algorithms
using the first and/or second derivatives information might be preferred
for their better performances and robustness in general.
Method *Powell* is a modification of Powell's method [3]_, [4]_ which
is a conjugate direction method. It performs sequential one-dimensional
minimizations along each vector of the directions set (`direc` field in
`options` and `info`), which is updated at each iteration of the main
minimization loop. The function need not be differentiable, and no
derivatives are taken.
Method *CG* uses a nonlinear conjugate gradient algorithm by Polak and
Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.
120-122. Only the first derivatives are used.
Method *BFGS* uses the quasi-Newton method of Broyden, Fletcher,
Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives
only. BFGS has proven good performance even for non-smooth
optimizations
Method *Newton-CG* uses a Newton-CG algorithm [5]_ pp. 168 (also known
as the truncated Newton method). It uses a CG method to the compute the
search direction. See also `fmin_tnc` for a box-constrained
minimization with a similar algorithm.
Method *Anneal* uses simulated annealing, which is a probabilistic
metaheuristic algorithm for global optimization. It uses no derivative
information from the function being optimized.
References
----------
.. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function
Minimization. The Computer Journal 7: 308-13.
.. [2] Wright M H. 1996. Direct search methods: Once scorned, now
respectable, in Numerical Analysis 1995: Proceedings of the 1995
Dundee Biennial Conference in Numerical Analysis (Eds. D F
Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.
191-208.
.. [3] Powell, M J D. 1964. An efficient method for finding the minimum of
a function of several variables without calculating derivatives. The
Computer Journal 7: 155-162.
.. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery.
Numerical Recipes (any edition), Cambridge University Press.
.. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.
Springer New York.
Examples
--------
Let us consider the problem of minimizing the Rosenbrock function. This
function (and its respective derivatives) is implemented in `rosen`
(resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`.
>>> from scipy.optimize import minimize, rosen, rosen_der
A simple application of the *Nelder-Mead* method is:
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = minimize(rosen, x0, method='Nelder-Mead')
Optimization terminated successfully.
Current function value: 0.000066
Iterations: 141
Function evaluations: 243
>>> print xopt
[ 1. 1. 1. 1. 1.]
Now using the *BFGS* algorithm, using the first derivative and a few
options:
>>> xopt, info = minimize(rosen, x0, method='BFGS', jac=rosen_der,
... options={'gtol': 1e-6, 'disp': False},
... full_output=True)
>>> print info['message']
Optimization terminated successfully.
>>> print info['solution']
[ 1. 1. 1. 1. 1.]
>>> print info['hess']
[[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377]
[ 0.01255155 0.02510441 0.04794055 0.09502834 0.18996269]
[ 0.02396251 0.04794055 0.09631614 0.19092151 0.38165151]
[ 0.04750988 0.09502834 0.19092151 0.38341252 0.7664427 ]
[ 0.09495377 0.18996269 0.38165151 0.7664427 1.53713523]]
"""
if method.lower() == 'nelder-mead':
return _minimize_neldermead(fun, x0, args, options, full_output,
retall, callback)
elif method.lower() == 'powell':
return _minimize_powell(fun, x0, args, options, full_output,
retall, callback)
elif method.lower() == 'cg':
return _minimize_cg(fun, x0, args, jac, options, full_output,
retall, callback)
elif method.lower() == 'bfgs':
return _minimize_bfgs(fun, x0, args, jac, options, full_output,
retall, callback)
elif method.lower() == 'newton-cg':
return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, options,
full_output, retall, callback)
elif method.lower() == 'anneal':
if callback:
warn("Method 'Anneal' does not support callback.",
RuntimeWarning)
if retall:
warn("Method 'Anneal' does not support retall option.",
RuntimeWarning)
return _minimize_anneal(fun, x0, args, options, full_output)
else:
raise ValueError('Unknown solver %s' % method)
def minimize(fun, x0, args=(), method='BFGS', jac=None, hess=None,
hessp=None, bounds=None, constraints=(),
options=dict(), full_output=False, callback=None,
retall=False):
"""
Minimization of scalar function of one or more variables.
Parameters
----------
fun : callable
Objective function.
x0 : ndarray
Initial guess.
args : tuple, optional
Extra arguments passed to the objective function and its
derivatives (Jacobian, Hessian).
method : str, optional
Type of solver. Should be one of:
{'Nelder-Mead', 'Powell', 'CG', 'BFGS', 'Newton-CG', 'Anneal',
'L-BFGS-B', 'TNC', 'COBYLA', 'SLSQP'}.
jac : bool or callable, optional
Jacobian of objective function. Only for CG, BFGS, Newton-CG.
If `jac` is a Boolean and is True, `fun` is assumed to return the
value of Jacobian along with the objective function. If False, the
Jacobian will be estimated numerically.
`jac` can also be a callable returning the Jacobian of the
objective. In this case, it must accept the same arguments as
`fun`.
hess, hessp : callable, optional
Hessian of objective function or Hessian of objective function
times an arbitrary vector p. Only for Newton-CG.
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.
bounds : sequence, optional
Bounds for variables (only for L-BFGS-B, TNC, COBYLA and SLSQP).
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None for one of ``min`` or
``max`` when there is no bound in that direction.
constraints : dict or sequence of dict, optional
Constraints definition (only for COBYLA and SLSQP).
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.
options : dict, optional
A dictionary of solver options. All methods accept the following
generic options:
maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.
For method-specific options, see `show_minimize_options`.
full_output : bool, optional
If True, return optional outputs. Default is False.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
retall : bool, optional
If True, return a list of the solution at each iteration. This is only
done if `full_output` is True.
Returns
-------
xopt : ndarray
The solution.
info : dict
A dictionary of optional outputs (depending on the chosen method)
with the keys:
solution : ndarray
The solution (same as `xopt`).
success : bool
Boolean flag indicating if a solution was found.
status : int
An integer flag indicating the type of termination. Its
value depends on the underlying solver. Refer to `message`
for more information.
message : str
A string message giving information about the cause of the
termination.
fun, jac, hess : ndarray
Values of objective function, Jacobian and Hessian (if
available).
nfev, njev, nhev: int
Number of evaluations of the objective functions and of its
jacobian and hessian.
nit: int
Number of iterations.
direc: ndarray
Current set of direction vectors for the Powell method.
T : float
Final temperature for simulated annealing.
accept : int
Number of tests accepted.
allvecs : list
Solution at each iteration (if ``retall == True``).
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter. The default method is *BFGS*.
Unconstrained minimization
~~~~~~~~~~~~~~~~~~~~~~~~~~
Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This
algorithm has been successful in many applications but other algorithms
using the first and/or second derivatives information might be preferred
for their better performances and robustness in general.
Method *Powell* is a modification of Powell's method [3]_, [4]_ which
is a conjugate direction method. It performs sequential one-dimensional
minimizations along each vector of the directions set (`direc` field in
`options` and `info`), which is updated at each iteration of the main
minimization loop. The function need not be differentiable, and no
derivatives are taken.
Method *CG* uses a nonlinear conjugate gradient algorithm by Polak and
Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.
120-122. Only the first derivatives are used.
Method *BFGS* uses the quasi-Newton method of Broyden, Fletcher,
Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives
only. BFGS has proven good performance even for non-smooth
optimizations
Method *Newton-CG* uses a Newton-CG algorithm [5]_ pp. 168 (also known
as the truncated Newton method). It uses a CG method to the compute the
search direction. See also *TNC* method for a box-constrained
minimization with a similar algorithm.
Method *Anneal* uses simulated annealing, which is a probabilistic
metaheuristic algorithm for global optimization. It uses no derivative
information from the function being optimized.
Constrained minimization
~~~~~~~~~~~~~~~~~~~~~~~~
Method *L-BFGS-B* uses the L-BFGS-B algorithm [6]_, [7]_ for bound
constrained minimization.
Method *TNC* uses a truncated Newton algorithm [5]_, [8]_ to minimize a
function with variables subject to bounds. This algorithm is uses
gradient information; it is also called Newton Conjugate-Gradient. It
differs from the *Newton-CG* method described above as it wraps a C
implementation and allows each variable to be given upper and lower
bounds.
Method *COBYLA* uses the Constrained Optimization BY Linear
Approximation (COBYLA) method [9]_, [10]_, [11]_. The algorithm is
based on linear approximations to the objective function and each
constraint. The method wraps a FORTRAN implementation of the algorithm.
Method *SLSQP* uses Sequential Least SQuares Programming to minimize a
function of several variables with any combination of bounds, equality
and inequality constraints. The method wraps the SLSQP Optimization
subroutine originally implemented by Dieter Kraft [12]_.
References
----------
.. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function
Minimization. The Computer Journal 7: 308-13.
.. [2] Wright M H. 1996. Direct search methods: Once scorned, now
respectable, in Numerical Analysis 1995: Proceedings of the 1995
Dundee Biennial Conference in Numerical Analysis (Eds. D F
Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.
191-208.
.. [3] Powell, M J D. 1964. An efficient method for finding the minimum of
a function of several variables without calculating derivatives. The
Computer Journal 7: 155-162.
.. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery.
Numerical Recipes (any edition), Cambridge University Press.
.. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.
Springer New York.
.. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory
Algorithm for Bound Constrained Optimization. SIAM Journal on
Scientific and Statistical Computing 16 (5): 1190-1208.
.. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm
778: L-BFGS-B, FORTRAN routines for large scale bound constrained
optimization. ACM Transactions on Mathematical Software 23 (4):
550-560.
.. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method.
1984. SIAM Journal of Numerical Analysis 21: 770-778.
.. [9] Powell, M J D. A direct search optimization method that models
the objective and constraint functions by linear interpolation.
1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez
and J-P Hennart, Kluwer Academic (Dordrecht), 51-67.
.. [10] Powell M J D. Direct search algorithms for optimization
calculations. 1998. Acta Numerica 7: 287-336.
.. [11] Powell M J D. A view of algorithms for optimization without
derivatives. 2007.Cambridge University Technical Report DAMTP
2007/NA03
.. [12] Kraft, D. A software package for sequential quadratic
programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace
Center -- Institute for Flight Mechanics, Koln, Germany.
Examples
--------
Let us consider the problem of minimizing the Rosenbrock function. This
function (and its respective derivatives) is implemented in `rosen`
(resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`.
>>> from scipy.optimize import minimize, rosen, rosen_der
A simple application of the *Nelder-Mead* method is:
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = minimize(rosen, x0, method='Nelder-Mead')
Optimization terminated successfully.
Current function value: 0.000066
Iterations: 141
Function evaluations: 243
>>> print xopt
[ 1. 1. 1. 1. 1.]
Now using the *BFGS* algorithm, using the first derivative and a few
options:
>>> xopt, info = minimize(rosen, x0, method='BFGS', jac=rosen_der,
... options={'gtol': 1e-6, 'disp': False},
... full_output=True)
>>> print info['message']
Optimization terminated successfully.
>>> print info['solution']
[ 1. 1. 1. 1. 1.]
>>> print info['hess']
[[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377]
[ 0.01255155 0.02510441 0.04794055 0.09502834 0.18996269]
[ 0.02396251 0.04794055 0.09631614 0.19092151 0.38165151]
[ 0.04750988 0.09502834 0.19092151 0.38341252 0.7664427 ]
[ 0.09495377 0.18996269 0.38165151 0.7664427 1.53713523]]
Next, consider a minimization problem with several constraints (namely
Example 16.4 from [5]_). The objective function is:
>>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2
There are three constraints defined as:
>>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2},
... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6},
... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2})
And variables must be positive, hence the following bounds:
>>> bnds = ((0, None), (0, None))
The optimization problem is solved using the SLSQP method as:
>>> xopt, info = minimize(fun, (2, 0), method='SLSQP', bounds=bnds,
... constraints=cons, full_output=True)
It should converge to the theoretical solution (1.4 ,1.7).
"""
meth = method.lower()
# check if optional parameters are supported by the selected method
# - jac
if meth in ['nelder-mead', 'powell', 'anneal', 'cobyla'] and bool(jac):
warn('Method %s does not use gradient information (jac).' % method,
RuntimeWarning)
# - hess
if meth != 'newton-cg' and hess is not None:
warn('Method %s does not use Hessian information (hess).' % method,
RuntimeWarning)
# - constraints or bounds
if meth in ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg'] and \
(bounds is not None or any(constraints)):
warn('Method %s cannot handle constraints nor bounds.' % method,
RuntimeWarning)
if meth in ['l-bfgs-b', 'tnc'] and any(constraints):
warn('Method %s cannot handle constraints.' % method,
RuntimeWarning)
if meth is 'cobyla' and bounds is not None:
warn('Method %s cannot handle bounds.' % method,
RuntimeWarning)
# - callback
if meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and \
callback is not None:
warn('Method %s does not support callback.' % method,
RuntimeWarning)
# - retall
if meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and \
retall:
warn('Method %s does not support retall.' % method,
RuntimeWarning)
# fun also returns the jacobian
if not callable(jac):
if bool(jac):
fun = MemoizeJac(fun)
jac = fun.derivative
else:
jac = None
if meth == 'nelder-mead':
return _minimize_neldermead(fun, x0, args, options, full_output,
retall, callback)
elif meth == 'powell':
return _minimize_powell(fun, x0, args, options, full_output,
retall, callback)
elif meth == 'cg':
return _minimize_cg(fun, x0, args, jac, options, full_output,
retall, callback)
elif meth == 'bfgs':
return _minimize_bfgs(fun, x0, args, jac, options, full_output,
retall, callback)
elif meth == 'newton-cg':
return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, options,
full_output, retall, callback)
elif meth == 'anneal':
return _minimize_anneal(fun, x0, args, options, full_output)
elif meth == 'l-bfgs-b':
return _minimize_lbfgsb(fun, x0, args, jac, bounds, options,
full_output)
elif meth == 'tnc':
return _minimize_tnc(fun, x0, args, jac, bounds, options,
full_output)
elif meth == 'cobyla':
return _minimize_cobyla(fun, x0, args, constraints, options,
full_output)
elif meth == 'slsqp':
return _minimize_slsqp(fun, x0, args, jac, bounds,
constraints, options, full_output)
else:
raise ValueError('Unknown solver %s' % method)
def minimize(fun,
x0,
args=(),
method='Nelder-Mead',
jac=None,
hess=None,
hessp=None,
options=dict(),
full_output=False,
callback=None,
retall=False):
"""
Minimization of scalar function of one or more variables.
Parameters
----------
fun : callable
Objective function.
x0 : ndarray
Initial guess.
args : tuple, optional
Extra arguments passed to the objective function and its
derivatives (Jacobian, Hessian).
method : str, optional
Type of solver. Should be one of:
{'Nelder-Mead', 'Powell', 'CG', 'BFGS', 'Newton-CG', 'Anneal'}.
jac : callable, optional
Jacobian of objective function (if None, Jacobian will be
estimated numerically). Only for CG, BFGS, Newton-CG.
hess, hessp : callable, optional
Hessian of objective function or Hessian of objective function
times an arbitrary vector p. Only for Newton-CG.
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. If it is not given, finite-differences
on `jac` are used to compute it.
options : dict, optional
A dictionary of solver options. All methods accept the following
generic options:
maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.
For method-specific options, see `show_minimize_options`.
full_output : bool, optional
If True, return optional outputs. Default is False.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
retall : bool, optional
If True, return a list of the solution at each iteration. This is only
done if `full_output` is True.
Returns
-------
xopt : ndarray
The solution.
info : dict
A dictionary of optional outputs (depending on the chosen method)
with the keys:
solution : ndarray
The solution (same as `xopt`).
success : bool
Boolean flag indicating if a solution was found.
status : int
An integer flag indicating the type of termination. Its
value depends on the underlying solver. Refer to `message`
for more information.
message : str
A string message giving information about the cause of the
termination.
fun, jac, hess : ndarray
Values of objective function, Jacobian and Hessian (if
available).
nfev, njev, nhev: int
Number of evaluations of the objective functions and of its
jacobian and hessian.
nit: int
Number of iterations.
direc: ndarray
Current set of direction vectors for the Powell method.
T : float
Final temperature for simulated annealing.
accept : int
Number of tests accepted.
allvecs : list
Solution at each iteration (if ``retall == True``).
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter.
Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This
algorithm has been successful in many applications but other algorithms
using the first and/or second derivatives information might be preferred
for their better performances and robustness in general.
Method *Powell* is a modification of Powell's method [3]_, [4]_ which
is a conjugate direction method. It performs sequential one-dimensional
minimizations along each vector of the directions set (`direc` field in
`options` and `info`), which is updated at each iteration of the main
minimization loop. The function need not be differentiable, and no
derivatives are taken.
Method *CG* uses a nonlinear conjugate gradient algorithm by Polak and
Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.
120-122. Only the first derivatives are used.
Method *BFGS* uses the quasi-Newton method of Broyden, Fletcher,
Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives
only. BFGS has proven good performance even for non-smooth
optimizations
Method *Newton-CG* uses a Newton-CG algorithm [5]_ pp. 168 (also known
as the truncated Newton method). It uses a CG method to the compute the
search direction. See also `fmin_tnc` for a box-constrained
minimization with a similar algorithm.
Method *Anneal* uses simulated annealing, which is a probabilistic
metaheuristic algorithm for global optimization. It uses no derivative
information from the function being optimized.
References
----------
.. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function
Minimization. The Computer Journal 7: 308-13.
.. [2] Wright M H. 1996. Direct search methods: Once scorned, now
respectable, in Numerical Analysis 1995: Proceedings of the 1995
Dundee Biennial Conference in Numerical Analysis (Eds. D F
Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.
191-208.
.. [3] Powell, M J D. 1964. An efficient method for finding the minimum of
a function of several variables without calculating derivatives. The
Computer Journal 7: 155-162.
.. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery.
Numerical Recipes (any edition), Cambridge University Press.
.. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.
Springer New York.
Examples
--------
Let us consider the problem of minimizing the Rosenbrock function. This
function (and its respective derivatives) is implemented in `rosen`
(resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`.
>>> from scipy.optimize import minimize, rosen, rosen_der
A simple application of the *Nelder-Mead* method is:
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = minimize(rosen, x0, method='Nelder-Mead')
Optimization terminated successfully.
Current function value: 0.000066
Iterations: 141
Function evaluations: 243
>>> print xopt
[ 1. 1. 1. 1. 1.]
Now using the *BFGS* algorithm, using the first derivative and a few
options:
>>> xopt, info = minimize(rosen, x0, method='BFGS', jac=rosen_der,
... options={'gtol': 1e-6, 'disp': False},
... full_output=True)
>>> print info['message']
Optimization terminated successfully.
>>> print info['solution']
[ 1. 1. 1. 1. 1.]
>>> print info['hess']
[[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377]
[ 0.01255155 0.02510441 0.04794055 0.09502834 0.18996269]
[ 0.02396251 0.04794055 0.09631614 0.19092151 0.38165151]
[ 0.04750988 0.09502834 0.19092151 0.38341252 0.7664427 ]
[ 0.09495377 0.18996269 0.38165151 0.7664427 1.53713523]]
"""
if method.lower() == 'nelder-mead':
return _minimize_neldermead(fun, x0, args, options, full_output,
retall, callback)
elif method.lower() == 'powell':
return _minimize_powell(fun, x0, args, options, full_output, retall,
callback)
elif method.lower() == 'cg':
return _minimize_cg(fun, x0, args, jac, options, full_output, retall,
callback)
elif method.lower() == 'bfgs':
return _minimize_bfgs(fun, x0, args, jac, options, full_output, retall,
callback)
elif method.lower() == 'newton-cg':
return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, options,
full_output, retall, callback)
elif method.lower() == 'anneal':
if callback:
warn("Method 'Anneal' does not support callback.", RuntimeWarning)
if retall:
warn("Method 'Anneal' does not support retall option.",
RuntimeWarning)
return _minimize_anneal(fun, x0, args, options, full_output)
else:
raise ValueError('Unknown solver %s' % method)
def minimize(fun, x0, args=(), method='BFGS', jac=None, hess=None,
hessp=None, bounds=None, constraints=(), tol=None,
callback=None, options=None):
"""
Minimization of scalar function of one or more variables.
.. versionadded:: 0.11.0
Parameters
----------
fun : callable
Objective function.
x0 : ndarray
Initial guess.
args : tuple, optional
Extra arguments passed to the objective function and its
derivatives (Jacobian, Hessian).
method : str, optional
Type of solver. Should be one of
- 'Nelder-Mead'
- 'Powell'
- 'CG'
- 'BFGS'
- 'Newton-CG'
- 'Anneal'
- 'L-BFGS-B'
- 'TNC'
- 'COBYLA'
- 'SLSQP'
jac : bool or callable, optional
Jacobian of objective function. Only for CG, BFGS, Newton-CG.
If `jac` is a Boolean and is True, `fun` is assumed to return the
value of Jacobian along with the objective function. If False, the
Jacobian will be estimated numerically.
`jac` can also be a callable returning the Jacobian of the
objective. In this case, it must accept the same arguments as `fun`.
hess, hessp : callable, optional
Hessian of objective function or Hessian of objective function
times an arbitrary vector p. Only for Newton-CG.
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.
bounds : sequence, optional
Bounds for variables (only for L-BFGS-B, TNC, COBYLA and SLSQP).
``(min, max)`` pairs for each element in ``x``, defining
the bounds on that parameter. Use None for one of ``min`` or
``max`` when there is no bound in that direction.
constraints : dict or sequence of dict, optional
Constraints definition (only for COBYLA and SLSQP).
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.
tol : float, optional
Tolerance for termination. For detailed control, use solver-specific
options.
options : dict, optional
A dictionary of solver options. All methods accept the following
generic options:
maxiter : int
Maximum number of iterations to perform.
disp : bool
Set to True to print convergence messages.
For method-specific options, see `show_options('minimize', method)`.
callback : callable, optional
Called after each iteration, as ``callback(xk)``, where ``xk`` is the
current parameter vector.
Returns
-------
res : Result
The optimization result represented as a ``Result`` object.
Important attributes are: ``x`` the solution array, ``success`` a
Boolean flag indicating if the optimizer exited successfully and
``message`` which describes the cause of the termination. See
`Result` for a description of other attributes.
See also
--------
minimize_scalar: Interface to minimization algorithms for scalar
univariate functions.
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter. The default method is *BFGS*.
**Unconstrained minimization**
Method *Nelder-Mead* uses the Simplex algorithm [1]_, [2]_. This
algorithm has been successful in many applications but other algorithms
using the first and/or second derivatives information might be preferred
for their better performances and robustness in general.
Method *Powell* is a modification of Powell's method [3]_, [4]_ which
is a conjugate direction method. It performs sequential one-dimensional
minimizations along each vector of the directions set (`direc` field in
`options` and `info`), which is updated at each iteration of the main
minimization loop. The function need not be differentiable, and no
derivatives are taken.
Method *CG* uses a nonlinear conjugate gradient algorithm by Polak and
Ribiere, a variant of the Fletcher-Reeves method described in [5]_ pp.
120-122. Only the first derivatives are used.
Method *BFGS* uses the quasi-Newton method of Broyden, Fletcher,
Goldfarb, and Shanno (BFGS) [5]_ pp. 136. It uses the first derivatives
only. BFGS has proven good performance even for non-smooth
optimizations
Method *Newton-CG* uses a Newton-CG algorithm [5]_ pp. 168 (also known
as the truncated Newton method). It uses a CG method to the compute the
search direction. See also *TNC* method for a box-constrained
minimization with a similar algorithm.
Method *Anneal* uses simulated annealing, which is a probabilistic
metaheuristic algorithm for global optimization. It uses no derivative
information from the function being optimized.
**Constrained minimization**
Method *L-BFGS-B* uses the L-BFGS-B algorithm [6]_, [7]_ for bound
constrained minimization.
Method *TNC* uses a truncated Newton algorithm [5]_, [8]_ to minimize a
function with variables subject to bounds. This algorithm is uses
gradient information; it is also called Newton Conjugate-Gradient. It
differs from the *Newton-CG* method described above as it wraps a C
implementation and allows each variable to be given upper and lower
bounds.
Method *COBYLA* uses the Constrained Optimization BY Linear
Approximation (COBYLA) method [9]_, [10]_, [11]_. The algorithm is
based on linear approximations to the objective function and each
constraint. The method wraps a FORTRAN implementation of the algorithm.
Method *SLSQP* uses Sequential Least SQuares Programming to minimize a
function of several variables with any combination of bounds, equality
and inequality constraints. The method wraps the SLSQP Optimization
subroutine originally implemented by Dieter Kraft [12]_.
References
----------
.. [1] Nelder, J A, and R Mead. 1965. A Simplex Method for Function
Minimization. The Computer Journal 7: 308-13.
.. [2] Wright M H. 1996. Direct search methods: Once scorned, now
respectable, in Numerical Analysis 1995: Proceedings of the 1995
Dundee Biennial Conference in Numerical Analysis (Eds. D F
Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK.
191-208.
.. [3] Powell, M J D. 1964. An efficient method for finding the minimum of
a function of several variables without calculating derivatives. The
Computer Journal 7: 155-162.
.. [4] Press W, S A Teukolsky, W T Vetterling and B P Flannery.
Numerical Recipes (any edition), Cambridge University Press.
.. [5] Nocedal, J, and S J Wright. 2006. Numerical Optimization.
Springer New York.
.. [6] Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory
Algorithm for Bound Constrained Optimization. SIAM Journal on
Scientific and Statistical Computing 16 (5): 1190-1208.
.. [7] Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm
778: L-BFGS-B, FORTRAN routines for large scale bound constrained
optimization. ACM Transactions on Mathematical Software 23 (4):
550-560.
.. [8] Nash, S G. Newton-Type Minimization Via the Lanczos Method.
1984. SIAM Journal of Numerical Analysis 21: 770-778.
.. [9] Powell, M J D. A direct search optimization method that models
the objective and constraint functions by linear interpolation.
1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez
and J-P Hennart, Kluwer Academic (Dordrecht), 51-67.
.. [10] Powell M J D. Direct search algorithms for optimization
calculations. 1998. Acta Numerica 7: 287-336.
.. [11] Powell M J D. A view of algorithms for optimization without
derivatives. 2007.Cambridge University Technical Report DAMTP
2007/NA03
.. [12] Kraft, D. A software package for sequential quadratic
programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace
Center -- Institute for Flight Mechanics, Koln, Germany.
Examples
--------
Let us consider the problem of minimizing the Rosenbrock function. This
function (and its respective derivatives) is implemented in `rosen`
(resp. `rosen_der`, `rosen_hess`) in the `scipy.optimize`.
>>> from scipy.optimize import minimize, rosen, rosen_der
A simple application of the *Nelder-Mead* method is:
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> res = minimize(rosen, x0, method='Nelder-Mead')
>>> res.x
[ 1. 1. 1. 1. 1.]
Now using the *BFGS* algorithm, using the first derivative and a few
options:
>>> res = minimize(rosen, x0, method='BFGS', jac=rosen_der,
... options={'gtol': 1e-6, 'disp': True})
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 52
Function evaluations: 64
Gradient evaluations: 64
>>> res.x
[ 1. 1. 1. 1. 1.]
>>> print res.message
Optimization terminated successfully.
>>> res.hess
[[ 0.00749589 0.01255155 0.02396251 0.04750988 0.09495377]
[ 0.01255155 0.02510441 0.04794055 0.09502834 0.18996269]
[ 0.02396251 0.04794055 0.09631614 0.19092151 0.38165151]
[ 0.04750988 0.09502834 0.19092151 0.38341252 0.7664427 ]
[ 0.09495377 0.18996269 0.38165151 0.7664427 1.53713523]]
Next, consider a minimization problem with several constraints (namely
Example 16.4 from [5]_). The objective function is:
>>> fun = lambda x: (x[0] - 1)**2 + (x[1] - 2.5)**2
There are three constraints defined as:
>>> cons = ({'type': 'ineq', 'fun': lambda x: x[0] - 2 * x[1] + 2},
... {'type': 'ineq', 'fun': lambda x: -x[0] - 2 * x[1] + 6},
... {'type': 'ineq', 'fun': lambda x: -x[0] + 2 * x[1] + 2})
And variables must be positive, hence the following bounds:
>>> bnds = ((0, None), (0, None))
The optimization problem is solved using the SLSQP method as:
>>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds,
... constraints=cons)
It should converge to the theoretical solution (1.4 ,1.7).
"""
meth = method.lower()
if options is None:
options = {}
# check if optional parameters are supported by the selected method
# - jac
if meth in ['nelder-mead', 'powell', 'anneal', 'cobyla'] and bool(jac):
warn('Method %s does not use gradient information (jac).' % method,
RuntimeWarning)
# - hess
if meth != 'newton-cg' and hess is not None:
warn('Method %s does not use Hessian information (hess).' % method,
RuntimeWarning)
# - constraints or bounds
if (meth in ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg'] and
(bounds is not None or any(constraints))):
warn('Method %s cannot handle constraints nor bounds.' % method,
RuntimeWarning)
if meth in ['l-bfgs-b', 'tnc'] and any(constraints):
warn('Method %s cannot handle constraints.' % method,
RuntimeWarning)
if meth is 'cobyla' and bounds is not None:
warn('Method %s cannot handle bounds.' % method,
RuntimeWarning)
# - callback
if (meth in ['anneal', 'tnc', 'cobyla', 'slsqp'] and
callback is not None):
warn('Method %s does not support callback.' % method,
RuntimeWarning)
# - return_all
if (meth in ['anneal', 'l-bfgs-b', 'tnc', 'cobyla', 'slsqp'] and
options.get('return_all', False)):
warn('Method %s does not support the return_all option.' % method,
RuntimeWarning)
# fun also returns the jacobian
if not callable(jac):
if bool(jac):
fun = MemoizeJac(fun)
jac = fun.derivative
else:
jac = None
# set default tolerances
if tol is not None:
options = dict(options)
if meth in ['nelder-mead', 'newton-cg', 'powell', 'tnc']:
options.setdefault('xtol', tol)
if meth in ['nelder-mead', 'powell', 'anneal', 'l-bfgs-b', 'tnc',
'slsqp']:
options.setdefault('ftol', tol)
if meth in ['bfgs', 'cg', 'l-bfgs-b', 'tnc']:
options.setdefault('gtol', tol)
if meth in ['cobyla']:
options.setdefault('tol', tol)
if meth == 'nelder-mead':
return _minimize_neldermead(fun, x0, args, callback, **options)
elif meth == 'powell':
return _minimize_powell(fun, x0, args, callback, **options)
elif meth == 'cg':
return _minimize_cg(fun, x0, args, jac, callback, **options)
elif meth == 'bfgs':
return _minimize_bfgs(fun, x0, args, jac, callback, **options)
elif meth == 'newton-cg':
return _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,
**options)
elif meth == 'anneal':
return _minimize_anneal(fun, x0, args, **options)
elif meth == 'l-bfgs-b':
return _minimize_lbfgsb(fun, x0, args, jac, bounds, callback=callback,
**options)
elif meth == 'tnc':
return _minimize_tnc(fun, x0, args, jac, bounds, **options)
elif meth == 'cobyla':
return _minimize_cobyla(fun, x0, args, constraints, **options)
elif meth == 'slsqp':
return _minimize_slsqp(fun, x0, args, jac, bounds,
constraints, **options)
else:
raise ValueError('Unknown solver %s' % method)
