_minimize.py

"""
Unified interfaces to minimization algorithms.

Functions
---------
- minimize : minimization of a function of several variables.
- minimize_scalar : minimization of a function of one variable.
"""
from __future__ import division, print_function, absolute_import


__all__ = ['minimize', 'minimize_scalar']


from warnings import warn

import numpy as np

from scipy._lib.six import callable

# unconstrained minimization
from .optimize import (_minimize_neldermead, _minimize_powell, _minimize_cg,
                      _minimize_bfgs, _minimize_newtoncg,
                      _minimize_scalar_brent, _minimize_scalar_bounded,
                      _minimize_scalar_golden, MemoizeJac)
from ._trustregion_dogleg import _minimize_dogleg
from ._trustregion_ncg import _minimize_trust_ncg

# constrained minimization
from .lbfgsb import _minimize_lbfgsb
from .tnc import _minimize_tnc
from .cobyla import _minimize_cobyla
from .slsqp import _minimize_slsqp


def minimize(fun, x0, args=(), method=None, jac=None, hess=None,
             hessp=None, bounds=None, constraints=(), tol=None,
             callback=None, options=None):
    """Minimization of scalar function of one or more variables.
    
    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.
    ``g_i(x)`` are the inequality constraints.
    ``h_j(x)`` are the equality constrains.

    Optionally, the lower and upper bounds for each element in x can also be
    specified using the `bounds` argument.

    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 or callable, optional
        Type of solver.  Should be one of

            - 'Nelder-Mead' :ref:`(see here) <optimize.minimize-neldermead>`
            - 'Powell'      :ref:`(see here) <optimize.minimize-powell>`
            - 'CG'          :ref:`(see here) <optimize.minimize-cg>`
            - 'BFGS'        :ref:`(see here) <optimize.minimize-bfgs>`
            - 'Newton-CG'   :ref:`(see here) <optimize.minimize-newtoncg>`
            - 'L-BFGS-B'    :ref:`(see here) <optimize.minimize-lbfgsb>`
            - 'TNC'         :ref:`(see here) <optimize.minimize-tnc>`
            - 'COBYLA'      :ref:`(see here) <optimize.minimize-cobyla>`
            - 'SLSQP'       :ref:`(see here) <optimize.minimize-slsqp>`
            - 'dogleg'      :ref:`(see here) <optimize.minimize-dogleg>`
            - 'trust-ncg'   :ref:`(see here) <optimize.minimize-trustncg>`
            - custom - a callable object (added in version 0.14.0),
              see below for description.

        If not given, chosen to be one of ``BFGS``, ``L-BFGS-B``, ``SLSQP``,
        depending if the problem has constraints or bounds.
    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`.
    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.
    bounds : sequence, optional
        Bounds for variables (only for L-BFGS-B, TNC 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 :func:`show_options()`.
    callback : callable, optional
        Called after each iteration, as ``callback(xk)``, where ``xk`` is the
        current parameter vector.

    Returns
    -------
    res : OptimizeResult
        The optimization result represented as a ``OptimizeResult`` 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
        `OptimizeResult` for a description of other attributes.


    See also
    --------
    minimize_scalar : Interface to minimization algorithms for scalar
        univariate functions
    show_options : Additional options accepted by the solvers

    Notes
    -----
    This section describes the available solvers that can be selected by the
    'method' parameter. The default method is *BFGS*.

    **Unconstrained minimization**

    Method :ref:`Nelder-Mead <optimize.minimize-neldermead>` uses the
    Simplex algorithm [1]_, [2]_. This algorithm is robust in many
    applications. However, if numerical computation of derivative can be
    trusted, other algorithms using the first and/or second derivatives
    information might be preferred for their better performance in
    general.

    Method :ref:`Powell <optimize.minimize-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 :ref:`CG <optimize.minimize-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 :ref:`BFGS <optimize.minimize-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. This method also
    returns an approximation of the Hessian inverse, stored as
    `hess_inv` in the OptimizeResult object.

    Method :ref:`Newton-CG <optimize.minimize-newtoncg>` 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 :ref:`dogleg <optimize.minimize-dogleg>` uses the dog-leg
    trust-region algorithm [5]_ for unconstrained minimization. This
    algorithm requires the gradient and Hessian; furthermore the
    Hessian is required to be positive definite.

    Method :ref:`trust-ncg <optimize.minimize-trustncg>` uses the
    Newton conjugate gradient trust-region algorithm [5]_ for
    unconstrained minimization. This algorithm requires the gradient
    and either the Hessian or a function that computes the product of
    the Hessian with a given vector.

    **Constrained minimization**

    Method :ref:`L-BFGS-B <optimize.minimize-lbfgsb>` uses the L-BFGS-B
    algorithm [6]_, [7]_ for bound constrained minimization.

    Method :ref:`TNC <optimize.minimize-tnc>` uses a truncated Newton
    algorithm [5]_, [8]_ to minimize a function with variables subject
    to bounds. This algorithm 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 :ref:`COBYLA <optimize.minimize-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. The
    constraints functions 'fun' may return either a single number
    or an array or list of numbers.

    Method :ref:`SLSQP <optimize.minimize-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]_. Note that the
    wrapper handles infinite values in bounds by converting them into
    large floating values.

    **Custom minimizers**

    It may be useful to pass a custom minimization method, for example
    when using a frontend to this method such as `scipy.optimize.basinhopping`
    or a different library.  You can simply pass a callable as the ``method``
    parameter.

    The callable is called as ``method(fun, x0, args, **kwargs, **options)``
    where ``kwargs`` corresponds to any other parameters passed to `minimize`
    (such as `callback`, `hess`, etc.), except the `options` dict, which has
    its contents also passed as `method` parameters pair by pair.  Also, if
    `jac` has been passed as a bool type, `jac` and `fun` are mangled so that
    `fun` returns just the function values and `jac` is converted to a function
    returning the Jacobian.  The method shall return an ``OptimizeResult``
    object.

    The provided `method` callable must be able to accept (and possibly ignore)
    arbitrary parameters; the set of parameters accepted by `minimize` may
    expand in future versions and then these parameters will be passed to
    the method.  You can find an example in the scipy.optimize tutorial.

    .. versionadded:: 0.11.0

    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', tol=1e-6)
    >>> res.x
    array([ 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: 26
             Function evaluations: 31
             Gradient evaluations: 31
    >>> res.x
    array([ 1.,  1.,  1.,  1.,  1.])
    >>> print(res.message)
    Optimization terminated successfully.
    >>> res.hess_inv
    array([[ 0.00749589,  0.01255155,  0.02396251,  0.04750988,  0.09495377],  # may vary
           [ 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).

    """
    x0 = np.asarray(x0)
    if x0.dtype.kind in np.typecodes["AllInteger"]:
        x0 = np.asarray(x0, dtype=float)

    if not isinstance(args, tuple):
        args = (args,)

    if method is None:
        # Select automatically
        if constraints:
            method = 'SLSQP'
        elif bounds is not None:
            method = 'L-BFGS-B'
        else:
            method = 'BFGS'

    if callable(method):
        meth = "_custom"
    else:
        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', 'cobyla'] and bool(jac):
        warn('Method %s does not use gradient information (jac).' % method,
             RuntimeWarning)
    # - hess
    if meth not in ('newton-cg', 'dogleg', 'trust-ncg', '_custom') and hess is not None:
        warn('Method %s does not use Hessian information (hess).' % method,
             RuntimeWarning)
    # - hessp
    if meth not in ('newton-cg', 'dogleg', 'trust-ncg', '_custom') and hessp is not None:
        warn('Method %s does not use Hessian-vector product '
                'information (hessp).' % method, RuntimeWarning)
    # - constraints or bounds
    if (meth in ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg', 'dogleg',
                 'trust-ncg'] and (bounds is not None or np.any(constraints))):
        warn('Method %s cannot handle constraints nor bounds.' % method,
             RuntimeWarning)
    if meth in ['l-bfgs-b', 'tnc'] and np.any(constraints):
        warn('Method %s cannot handle constraints.' % method,
             RuntimeWarning)
    if meth == 'cobyla' and bounds is not None:
        warn('Method %s cannot handle bounds.' % method,
             RuntimeWarning)
    # - callback
    if (meth in ['cobyla'] and callback is not None):
        warn('Method %s does not support callback.' % method, RuntimeWarning)
    # - return_all
    if (meth in ['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 == 'nelder-mead':
            options.setdefault('xatol', tol)
            options.setdefault('fatol', tol)
        if meth in ['newton-cg', 'powell', 'tnc']:
            options.setdefault('xtol', tol)
        if meth in ['powell', 'l-bfgs-b', 'tnc', 'slsqp']:
            options.setdefault('ftol', tol)
        if meth in ['bfgs', 'cg', 'l-bfgs-b', 'tnc', 'dogleg', 'trust-ncg']:
            options.setdefault('gtol', tol)
        if meth in ['cobyla', '_custom']:
            options.setdefault('tol', tol)

    if meth == '_custom':
        return method(fun, x0, args=args, jac=jac, hess=hess, hessp=hessp,
                      bounds=bounds, constraints=constraints,
                      callback=callback, **options)
    elif 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 == '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, callback=callback,
                             **options)
    elif meth == 'cobyla':
        return _minimize_cobyla(fun, x0, args, constraints, **options)
    elif meth == 'slsqp':
        return _minimize_slsqp(fun, x0, args, jac, bounds,
                               constraints, callback=callback, **options)
    elif meth == 'dogleg':
        return _minimize_dogleg(fun, x0, args, jac, hess,
                                callback=callback, **options)
    elif meth == 'trust-ncg':
        return _minimize_trust_ncg(fun, x0, args, jac, hess, hessp,
                                   callback=callback, **options)
    else:
        raise ValueError('Unknown solver %s' % method)


def minimize_scalar(fun, bracket=None, bounds=None, args=(),
                    method='brent', tol=None, options=None):
    """Minimization of scalar function of one variable.

    Parameters
    ----------
    fun : callable
        Objective function.
        Scalar function, must return a scalar.
    bracket : sequence, optional
        For methods 'brent' and 'golden', `bracket` defines the bracketing
        interval and can either have three items ``(a, b, c)`` so that
        ``a < b < c`` and ``fun(b) < fun(a), fun(c)`` or two items ``a`` and
        ``c`` which are assumed to be a starting interval for a downhill
        bracket search (see `bracket`); it doesn't always mean that the
        obtained solution will satisfy ``a <= x <= c``.
    bounds : sequence, optional
        For method 'bounded', `bounds` is mandatory and must have two items
        corresponding to the optimization bounds.
    args : tuple, optional
        Extra arguments passed to the objective function.
    method : str or callable, optional
        Type of solver.  Should be one of:

            - 'Brent'     :ref:`(see here) <optimize.minimize_scalar-brent>`
            - 'Bounded'   :ref:`(see here) <optimize.minimize_scalar-bounded>`
            - 'Golden'    :ref:`(see here) <optimize.minimize_scalar-golden>`
            - custom - a callable object (added in version 0.14.0), see below

    tol : float, optional
        Tolerance for termination. For detailed control, use solver-specific
        options.
    options : dict, optional
        A dictionary of solver options.

            maxiter : int
                Maximum number of iterations to perform.
            disp : bool
                Set to True to print convergence messages.

        See :func:`show_options()` for solver-specific options.

    Returns
    -------
    res : OptimizeResult
        The optimization result represented as a ``OptimizeResult`` 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
        `OptimizeResult` for a description of other attributes.

    See also
    --------
    minimize : Interface to minimization algorithms for scalar multivariate
        functions
    show_options : Additional options accepted by the solvers

    Notes
    -----
    This section describes the available solvers that can be selected by the
    'method' parameter. The default method is *Brent*.

    Method :ref:`Brent <optimize.minimize_scalar-brent>` uses Brent's
    algorithm to find a local minimum.  The algorithm uses inverse
    parabolic interpolation when possible to speed up convergence of
    the golden section method.

    Method :ref:`Golden <optimize.minimize_scalar-golden>` uses the
    golden section search technique. It uses analog of the bisection
    method to decrease the bracketed interval. It is usually
    preferable to use the *Brent* method.

    Method :ref:`Bounded <optimize.minimize_scalar-bounded>` can
    perform bounded minimization. It uses the Brent method to find a
    local minimum in the interval x1 < xopt < x2.

    **Custom minimizers**

    It may be useful to pass a custom minimization method, for example
    when using some library frontend to minimize_scalar.  You can simply
    pass a callable as the ``method`` parameter.

    The callable is called as ``method(fun, args, **kwargs, **options)``
    where ``kwargs`` corresponds to any other parameters passed to `minimize`
    (such as `bracket`, `tol`, etc.), except the `options` dict, which has
    its contents also passed as `method` parameters pair by pair.  The method
    shall return an ``OptimizeResult`` object.

    The provided `method` callable must be able to accept (and possibly ignore)
    arbitrary parameters; the set of parameters accepted by `minimize` may
    expand in future versions and then these parameters will be passed to
    the method.  You can find an example in the scipy.optimize tutorial.

    .. versionadded:: 0.11.0

    Examples
    --------
    Consider the problem of minimizing the following function.

    >>> def f(x):
    ...     return (x - 2) * x * (x + 2)**2

    Using the *Brent* method, we find the local minimum as:

    >>> from scipy.optimize import minimize_scalar
    >>> res = minimize_scalar(f)
    >>> res.x
    1.28077640403

    Using the *Bounded* method, we find a local minimum with specified
    bounds as:

    >>> res = minimize_scalar(f, bounds=(-3, -1), method='bounded')
    >>> res.x
    -2.0000002026

    """
    if not isinstance(args, tuple):
        args = (args,)

    if callable(method):
        meth = "_custom"
    else:
        meth = method.lower()
    if options is None:
        options = {}

    if tol is not None:
        options = dict(options)
        if meth == 'bounded' and 'xatol' not in options:
            warn("Method 'bounded' does not support relative tolerance in x; "
                 "defaulting to absolute tolerance.", RuntimeWarning)
            options['xatol'] = tol
        elif meth == '_custom':
            options.setdefault('tol', tol)
        else:
            options.setdefault('xtol', tol)

    if meth == '_custom':
        return method(fun, args=args, bracket=bracket, bounds=bounds, **options)
    elif meth == 'brent':
        return _minimize_scalar_brent(fun, bracket, args, **options)
    elif meth == 'bounded':
        if bounds is None:
            raise ValueError('The `bounds` parameter is mandatory for '
                             'method `bounded`.')
        return _minimize_scalar_bounded(fun, bounds, args, **options)
    elif meth == 'golden':
        return _minimize_scalar_golden(fun, bracket, args, **options)
    else:
        raise ValueError('Unknown solver %s' % method)