image-match/python/py/Lib/site-packages/scipy/optimize/_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.
"""

__all__ = ['minimize', 'minimize_scalar']


from warnings import warn

import numpy as np
from scipy._lib._util import wrapped_inspect_signature

# 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, OptimizeResult,
                        _wrap_callback, _recover_from_bracket_error)
from ._trustregion_dogleg import _minimize_dogleg
from ._trustregion_ncg import _minimize_trust_ncg
from ._trustregion_krylov import _minimize_trust_krylov
from ._trustregion_exact import _minimize_trustregion_exact
from ._trustregion_constr import _minimize_trustregion_constr

# constrained minimization
from ._lbfgsb_py import _minimize_lbfgsb
from ._tnc import _minimize_tnc
from ._cobyla_py import _minimize_cobyla
from ._cobyqa_py import _minimize_cobyqa
from ._slsqp_py import _minimize_slsqp
from ._constraints import (old_bound_to_new, new_bounds_to_old,
                           old_constraint_to_new, new_constraint_to_old,
                           NonlinearConstraint, LinearConstraint, Bounds,
                           PreparedConstraint)
from ._differentiable_functions import FD_METHODS

MINIMIZE_METHODS = ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg',
                    'l-bfgs-b', 'tnc', 'cobyla', 'cobyqa', 'slsqp',
                    'trust-constr', 'dogleg', 'trust-ncg', 'trust-exact',
                    'trust-krylov']

# These methods support the new callback interface (passed an OptimizeResult)
MINIMIZE_METHODS_NEW_CB = ['nelder-mead', 'powell', 'cg', 'bfgs', 'newton-cg',
                           'l-bfgs-b', 'trust-constr', 'dogleg', 'trust-ncg',
                           'trust-exact', 'trust-krylov', 'cobyqa', 'cobyla', 'slsqp']

MINIMIZE_SCALAR_METHODS = ['brent', 'bounded', 'golden']

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.

    Parameters
    ----------
    fun : callable
        The objective function to be minimized::

            fun(x, *args) -> float

        where ``x`` is a 1-D array with shape (n,) and ``args``
        is a tuple of the fixed parameters needed to completely
        specify the function.

        Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where
        ``my_args`` and ``my_kwargs`` are required positional and keyword arguments.
        Rather than passing ``f0`` as the callable, wrap it to accept
        only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the
        callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been
        gathered before invoking this function.
    x0 : ndarray, shape (n,)
        Initial guess. Array of real elements of size (n,),
        where ``n`` is the number of independent variables.
    args : tuple, optional
        Extra arguments passed to the objective function and its
        derivatives (`fun`, `jac` and `hess` functions).
    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>`
        - 'COBYQA'      :ref:`(see here) <optimize.minimize-cobyqa>`
        - 'SLSQP'       :ref:`(see here) <optimize.minimize-slsqp>`
        - 'trust-constr':ref:`(see here) <optimize.minimize-trustconstr>`
        - 'dogleg'      :ref:`(see here) <optimize.minimize-dogleg>`
        - 'trust-ncg'   :ref:`(see here) <optimize.minimize-trustncg>`
        - 'trust-exact' :ref:`(see here) <optimize.minimize-trustexact>`
        - 'trust-krylov' :ref:`(see here) <optimize.minimize-trustkrylov>`
        - custom - a callable object, see below for description.

        If not given, chosen to be one of ``BFGS``, ``L-BFGS-B``, ``SLSQP``,
        depending on whether or not the problem has constraints or bounds.
    jac : {callable,  '2-point', '3-point', 'cs', bool}, optional
        Method for computing the gradient vector. Only for CG, BFGS,
        Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg, trust-krylov,
        trust-exact and trust-constr.
        If it is a callable, it should be a function that returns the gradient
        vector::

            jac(x, *args) -> array_like, shape (n,)

        where ``x`` is an array with shape (n,) and ``args`` is a tuple with
        the fixed parameters. If `jac` is a Boolean and is True, `fun` is
        assumed to return a tuple ``(f, g)`` containing the objective
        function and the gradient.
        Methods 'Newton-CG', 'trust-ncg', 'dogleg', 'trust-exact', and
        'trust-krylov' require that either a callable be supplied, or that
        `fun` return the objective and gradient.
        If None or False, the gradient will be estimated using 2-point finite
        difference estimation with an absolute step size.
        Alternatively, the keywords  {'2-point', '3-point', 'cs'} can be used
        to select a finite difference scheme for numerical estimation of the
        gradient with a relative step size. These finite difference schemes
        obey any specified `bounds`.
    hess : {callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}, optional
        Method for computing the Hessian matrix. Only for Newton-CG, dogleg,
        trust-ncg, trust-krylov, trust-exact and trust-constr.
        If it is callable, it should return the Hessian matrix::

            hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)

        where ``x`` is a (n,) ndarray and ``args`` is a tuple with the fixed
        parameters.
        The keywords {'2-point', '3-point', 'cs'} can also be used to select
        a finite difference scheme for numerical estimation of the hessian.
        Alternatively, objects implementing the `HessianUpdateStrategy`
        interface can be used to approximate the Hessian. Available
        quasi-Newton methods implementing this interface are:

        - `BFGS`
        - `SR1`

        Not all of the options are available for each of the methods; for
        availability refer to the notes.
    hessp : callable, optional
        Hessian of objective function times an arbitrary vector p. Only for
        Newton-CG, trust-ncg, trust-krylov, trust-constr.
        Only one of `hessp` or `hess` needs to be given. If `hess` is
        provided, then `hessp` will be ignored. `hessp` must compute the
        Hessian times an arbitrary vector::

            hessp(x, p, *args) ->  ndarray shape (n,)

        where ``x`` is a (n,) ndarray, ``p`` is an arbitrary vector with
        dimension (n,) and ``args`` is a tuple with the fixed
        parameters.
    bounds : sequence or `Bounds`, optional
        Bounds on variables for Nelder-Mead, L-BFGS-B, TNC, SLSQP, Powell,
        trust-constr, COBYLA, and COBYQA methods. There are two ways to specify
        the bounds:

        1. Instance of `Bounds` class.
        2. Sequence of ``(min, max)`` pairs for each element in `x`. None
           is used to specify no bound.

    constraints : {Constraint, dict} or List of {Constraint, dict}, optional
        Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr.

        Constraints for 'trust-constr', 'cobyqa', and 'cobyla' are defined as a single
        object or a list of objects specifying constraints to the optimization problem.
        Available constraints are:

        - `LinearConstraint`
        - `NonlinearConstraint`

        Constraints for COBYLA, SLSQP are defined as a list of dictionaries.
        Each 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.

    tol : float, optional
        Tolerance for termination. When `tol` is specified, the selected
        minimization algorithm sets some relevant solver-specific tolerance(s)
        equal to `tol`. For detailed control, use solver-specific
        options.
    options : dict, optional
        A dictionary of solver options. All methods except `TNC` accept the
        following generic options:

        maxiter : int
            Maximum number of iterations to perform. Depending on the
            method each iteration may use several function evaluations.

            For `TNC` use `maxfun` instead of `maxiter`.
        disp : bool
            Set to True to print convergence messages.

        For method-specific options, see :func:`show_options()`.
    callback : callable, optional
        A callable called after each iteration.

        All methods except TNC support a callable with
        the signature::

            callback(intermediate_result: OptimizeResult)

        where ``intermediate_result`` is a keyword parameter containing an
        `OptimizeResult` with attributes ``x`` and ``fun``, the present values
        of the parameter vector and objective function. Not all attributes of
        `OptimizeResult` may be present. The name of the parameter must be
        ``intermediate_result`` for the callback to be passed an `OptimizeResult`.
        These methods will also terminate if the callback raises ``StopIteration``.

        All methods except trust-constr (also) support a signature like::

            callback(xk)

        where ``xk`` is the current parameter vector.

        Introspection is used to determine which of the signatures above to
        invoke.

    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:`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. Suitable for large-scale
    problems.

    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. Suitable for large-scale problems.

    Method :ref:`trust-krylov <optimize.minimize-trustkrylov>` uses
    the Newton GLTR trust-region algorithm [14]_, [15]_ 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. Suitable for large-scale problems.
    On indefinite problems it requires usually less iterations than the
    `trust-ncg` method and is recommended for medium and large-scale problems.

    Method :ref:`trust-exact <optimize.minimize-trustexact>`
    is a trust-region method for unconstrained minimization in which
    quadratic subproblems are solved almost exactly [13]_. This
    algorithm requires the gradient and the Hessian (which is
    *not* required to be positive definite). It is, in many
    situations, the Newton method to converge in fewer iterations
    and the most recommended for small and medium-size problems.

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

    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. If bounds are not provided, then an
    unbounded line search will be used. If bounds are provided and
    the initial guess is within the bounds, then every function
    evaluation throughout the minimization procedure will be within
    the bounds. If bounds are provided, the initial guess is outside
    the bounds, and `direc` is full rank (default has full rank), then
    some function evaluations during the first iteration may be
    outside the bounds, but every function evaluation after the first
    iteration will be within the bounds. If `direc` is not full rank,
    then some parameters may not be optimized and the solution is not
    guaranteed to be within the bounds.

    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.

    **Constrained Minimization**

    Method :ref:`COBYLA <optimize.minimize-cobyla>` uses the PRIMA
    implementation [19]_ of 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.

    Method :ref:`COBYQA <optimize.minimize-cobyqa>` uses the Constrained
    Optimization BY Quadratic Approximations (COBYQA) method [18]_. The
    algorithm is a derivative-free trust-region SQP method based on quadratic
    approximations to the objective function and each nonlinear constraint. The
    bounds are treated as unrelaxable constraints, in the sense that the
    algorithm always respects them throughout the optimization process.

    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.

    Method :ref:`trust-constr <optimize.minimize-trustconstr>` is a
    trust-region algorithm for constrained optimization. It switches
    between two implementations depending on the problem definition.
    It is the most versatile constrained minimization algorithm
    implemented in SciPy and the most appropriate for large-scale problems.
    For equality constrained problems it is an implementation of Byrd-Omojokun
    Trust-Region SQP method described in [17]_ and in [5]_, p. 549. When
    inequality constraints are imposed as well, it switches to the trust-region
    interior point method described in [16]_. This interior point algorithm,
    in turn, solves inequality constraints by introducing slack variables
    and solving a sequence of equality-constrained barrier problems
    for progressively smaller values of the barrier parameter.
    The previously described equality constrained SQP method is
    used to solve the subproblems with increasing levels of accuracy
    as the iterate gets closer to a solution.

    **Finite-Difference Options**

    For Method :ref:`trust-constr <optimize.minimize-trustconstr>`
    the gradient and the Hessian may be approximated using
    three finite-difference schemes: {'2-point', '3-point', 'cs'}.
    The scheme 'cs' is, potentially, the most accurate but it
    requires the function to correctly handle complex inputs and to
    be differentiable in the complex plane. The scheme '3-point' is more
    accurate than '2-point' but requires twice as many operations. If the
    gradient is estimated via finite-differences the Hessian must be
    estimated using one of the quasi-Newton strategies.

    **Method specific options for the** `hess` **keyword**

    +--------------+------+----------+-------------------------+-----+
    | method/Hess  | None | callable | '2-point/'3-point'/'cs' | HUS |
    +==============+======+==========+=========================+=====+
    | Newton-CG    | x    | (n, n)   | x                       | x   |
    |              |      | LO       |                         |     |
    +--------------+------+----------+-------------------------+-----+
    | dogleg       |      | (n, n)   |                         |     |
    +--------------+------+----------+-------------------------+-----+
    | trust-ncg    |      | (n, n)   | x                       | x   |
    +--------------+------+----------+-------------------------+-----+
    | trust-krylov |      | (n, n)   | x                       | x   |
    +--------------+------+----------+-------------------------+-----+
    | trust-exact  |      | (n, n)   |                         |     |
    +--------------+------+----------+-------------------------+-----+
    | trust-constr | x    | (n, n)   |  x                      | x   |
    |              |      | LO       |                         |     |
    |              |      | sp       |                         |     |
    +--------------+------+----------+-------------------------+-----+

    where LO=LinearOperator, sp=Sparse matrix, HUS=HessianUpdateStrategy

    **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.

    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.
    .. [13] Conn, A. R., Gould, N. I., and Toint, P. L.
       Trust region methods. 2000. Siam. pp. 169-200.
    .. [14] F. Lenders, C. Kirches, A. Potschka: "trlib: A vector-free
       implementation of the GLTR method for iterative solution of
       the trust region problem", :arxiv:`1611.04718`
    .. [15] N. Gould, S. Lucidi, M. Roma, P. Toint: "Solving the
       Trust-Region Subproblem using the Lanczos Method",
       SIAM J. Optim., 9(2), 504--525, (1999).
    .. [16] Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. 1999.
        An interior point algorithm for large-scale nonlinear  programming.
        SIAM Journal on Optimization 9.4: 877-900.
    .. [17] Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. 1998. On the
        implementation of an algorithm for large-scale equality constrained
        optimization. SIAM Journal on Optimization 8.3: 682-706.
    .. [18] Ragonneau, T. M. *Model-Based Derivative-Free Optimization Methods
        and Software*. PhD thesis, Department of Applied Mathematics, The Hong
        Kong Polytechnic University, Hong Kong, China, 2022. URL:
        https://theses.lib.polyu.edu.hk/handle/200/12294.
    .. [19] Zhang, Z. "PRIMA: Reference Implementation for Powell's Methods with
        Modernization and Amelioration", https://www.libprima.net,
        :doi:`10.5281/zenodo.8052654`
    .. [20] Karush-Kuhn-Tucker conditions,
        https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions

    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]``. *SLSQP* also
    returns the multipliers that are used in the solution of the problem. These
    multipliers, when the problem constraints are linear, can be thought of as the
    Karush-Kuhn-Tucker (KKT) multipliers, which are a generalization
    of Lagrange multipliers to inequality-constrained optimization problems ([20]_).

    Notice that at the solution, the first constraint is active. Let's evaluate the
    function at solution:

    >>> cons[0]['fun'](res.x)
    np.float64(1.4901224698604665e-09)

    Also, notice that at optimality there is a non-zero multiplier:

    >>> res.multipliers
    array([0.8, 0. , 0. ])

    This can be understood as the local sensitivity of the optimal value of the
    objective function with respect to changes in the first constraint. If we
    tighten the constraint by a small amount ``eps``:

    >>> eps = 0.01
    >>> cons[0]['fun'] = lambda x: x[0] - 2 * x[1] + 2 - eps

    we expect the optimal value of the objective function to increase by
    approximately ``eps * res.multipliers[0]``:

    >>> eps * res.multipliers[0]  # Expected change in f0
    np.float64(0.008000000027153205)
    >>> f0 = res.fun  # Keep track of the previous optimal value
    >>> res = minimize(fun, (2, 0), method='SLSQP', bounds=bnds, constraints=cons)
    >>> f1 = res.fun  # New optimal value
    >>> f1 - f0
    np.float64(0.008019998807885509)

    """
    x0 = np.atleast_1d(np.asarray(x0))

    if x0.ndim != 1:
        raise ValueError("'x0' must only have one dimension.")

    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', 'cobyqa') and bool(jac):
        warn(f'Method {method} does not use gradient information (jac).',
             RuntimeWarning, stacklevel=2)
    # - hess
    if meth not in ('newton-cg', 'dogleg', 'trust-ncg', 'trust-constr',
                    'trust-krylov', 'trust-exact', '_custom') and hess is not None:
        warn(f'Method {method} does not use Hessian information (hess).',
             RuntimeWarning, stacklevel=2)
    # - hessp
    if meth not in ('newton-cg', 'trust-ncg', 'trust-constr',
                    'trust-krylov', '_custom') \
       and hessp is not None:
        warn(f'Method {method} does not use Hessian-vector product'
             ' information (hessp).',
             RuntimeWarning, stacklevel=2)
    # - constraints or bounds
    if (meth not in ('cobyla', 'cobyqa', 'slsqp', 'trust-constr', '_custom') and
            np.any(constraints)):
        warn(f'Method {method} cannot handle constraints.',
             RuntimeWarning, stacklevel=2)
    if meth not in (
            'nelder-mead', 'powell', 'l-bfgs-b', 'cobyla', 'cobyqa', 'slsqp',
            'tnc', 'trust-constr', '_custom') and bounds is not None:
        warn(f'Method {method} cannot handle bounds.',
             RuntimeWarning, stacklevel=2)
    # - return_all
    if (meth in ('l-bfgs-b', 'tnc', 'cobyla', 'cobyqa', 'slsqp') and
            options.get('return_all', False)):
        warn(f'Method {method} does not support the return_all option.',
             RuntimeWarning, stacklevel=2)

    # check gradient vector
    if callable(jac):
        pass
    elif jac is True:
        # fun returns func and grad
        fun = MemoizeJac(fun)
        jac = fun.derivative
    elif (jac in FD_METHODS and
          meth in ['trust-constr', 'bfgs', 'cg', 'l-bfgs-b', 'tnc', 'slsqp']):
        # finite differences with relative step
        pass
    elif meth in ['trust-constr']:
        # default jac calculation for this method
        jac = '2-point'
    elif jac is None or bool(jac) is False:
        # this will cause e.g. LBFGS to use forward difference, absolute step
        jac = None
    else:
        # default if jac option is not understood
        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', 'trust-exact', 'trust-krylov'):
            options.setdefault('gtol', tol)
        if meth in ('cobyla', '_custom'):
            options.setdefault('tol', tol)
        if meth == 'cobyqa':
            options.setdefault('final_tr_radius', tol)
        if meth == 'trust-constr':
            options.setdefault('xtol', tol)
            options.setdefault('gtol', tol)
            options.setdefault('barrier_tol', tol)

    if meth == '_custom':
        # custom method called before bounds and constraints are 'standardised'
        # custom method should be able to accept whatever bounds/constraints
        # are provided to it.
        return method(fun, x0, args=args, jac=jac, hess=hess, hessp=hessp,
                      bounds=bounds, constraints=constraints,
                      callback=callback, **options)

    constraints = standardize_constraints(constraints, x0, meth)

    remove_vars = False
    if bounds is not None:
        # convert to new-style bounds so we only have to consider one case
        bounds = standardize_bounds(bounds, x0, 'new')
        bounds = _validate_bounds(bounds, x0, meth)

        if meth in {"tnc", "slsqp", "l-bfgs-b"}:
            # These methods can't take the finite-difference derivatives they
            # need when a variable is fixed by the bounds. To avoid this issue,
            # remove fixed variables from the problem.
            # NOTE: if this list is expanded, then be sure to update the
            # accompanying tests and test_optimize.eb_data. Consider also if
            # default OptimizeResult will need updating.

            # determine whether any variables are fixed
            i_fixed = (bounds.lb == bounds.ub)

            if np.all(i_fixed):
                # all the parameters are fixed, a minimizer is not able to do
                # anything
                return _optimize_result_for_equal_bounds(
                    fun, bounds, meth, args=args, constraints=constraints
                )

            # determine whether finite differences are needed for any grad/jac
            fd_needed = (not callable(jac))
            for con in constraints:
                if not callable(con.get('jac', None)):
                    fd_needed = True

            # If finite differences are ever used, remove all fixed variables
            # Always remove fixed variables for TNC; see gh-14565
            remove_vars = i_fixed.any() and (fd_needed or meth == "tnc")
            if remove_vars:
                x_fixed = (bounds.lb)[i_fixed]
                x0 = x0[~i_fixed]
                bounds = _remove_from_bounds(bounds, i_fixed)
                fun = _Remove_From_Func(fun, i_fixed, x_fixed)

                if callable(callback):
                    sig = wrapped_inspect_signature(callback)
                    if set(sig.parameters) == {'intermediate_result'}:
                        # callback(intermediate_result)
                        print(callback)
                        callback = _Patch_Callback_Equal_Variables(
                            callback, i_fixed, x_fixed
                        )
                    else:
                        # callback(x)
                        callback = _Remove_From_Func(callback, i_fixed, x_fixed)

                if callable(jac):
                    jac = _Remove_From_Func(jac, i_fixed, x_fixed, remove=1)

                # make a copy of the constraints so the user's version doesn't
                # get changed. (Shallow copy is ok)
                constraints = [con.copy() for con in constraints]
                for con in constraints:  # yes, guaranteed to be a list
                    con['fun'] = _Remove_From_Func(con['fun'], i_fixed,
                                                   x_fixed, min_dim=1,
                                                   remove=0)
                    if callable(con.get('jac', None)):
                        con['jac'] = _Remove_From_Func(con['jac'], i_fixed,
                                                       x_fixed, min_dim=2,
                                                       remove=1)
        bounds = standardize_bounds(bounds, x0, meth)

    # selects whether to use callback(x) or callback(intermediate_result)
    callback = _wrap_callback(callback, meth)

    if meth == 'nelder-mead':
        res = _minimize_neldermead(fun, x0, args, callback, bounds=bounds,
                                   **options)
    elif meth == 'powell':
        res = _minimize_powell(fun, x0, args, callback, bounds, **options)
    elif meth == 'cg':
        res = _minimize_cg(fun, x0, args, jac, callback, **options)
    elif meth == 'bfgs':
        res = _minimize_bfgs(fun, x0, args, jac, callback, **options)
    elif meth == 'newton-cg':
        res = _minimize_newtoncg(fun, x0, args, jac, hess, hessp, callback,
                                 **options)
    elif meth == 'l-bfgs-b':
        res = _minimize_lbfgsb(fun, x0, args, jac, bounds,
                               callback=callback, **options)
    elif meth == 'tnc':
        res = _minimize_tnc(fun, x0, args, jac, bounds, callback=callback,
                            **options)
    elif meth == 'cobyla':
        res = _minimize_cobyla(fun, x0, args, constraints, callback=callback,
                               bounds=bounds, **options)
    elif meth == 'cobyqa':
        res = _minimize_cobyqa(fun, x0, args, bounds, constraints, callback,
                               **options)
    elif meth == 'slsqp':
        res = _minimize_slsqp(fun, x0, args, jac, bounds,
                              constraints, callback=callback, **options)
    elif meth == 'trust-constr':
        res = _minimize_trustregion_constr(fun, x0, args, jac, hess, hessp,
                                           bounds, constraints,
                                           callback=callback, **options)
    elif meth == 'dogleg':
        res = _minimize_dogleg(fun, x0, args, jac, hess,
                               callback=callback, **options)
    elif meth == 'trust-ncg':
        res = _minimize_trust_ncg(fun, x0, args, jac, hess, hessp,
                                  callback=callback, **options)
    elif meth == 'trust-krylov':
        res = _minimize_trust_krylov(fun, x0, args, jac, hess, hessp,
                                     callback=callback, **options)
    elif meth == 'trust-exact':
        res = _minimize_trustregion_exact(fun, x0, args, jac, hess,
                                          callback=callback, **options)
    else:
        raise ValueError(f'Unknown solver {method}')

    if remove_vars:
        res.x = _add_to_array(res.x, i_fixed, x_fixed)
        res.jac = _add_to_array(res.jac, i_fixed, np.nan)
        if "hess_inv" in res:
            res.hess_inv = None  # unknown

    if getattr(callback, 'stop_iteration', False):
        res.success = False
        res.status = 99
        res.message = "`callback` raised `StopIteration`."

    return res


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

    Parameters
    ----------
    fun : callable
        Objective function.
        Scalar function, must return a scalar.

        Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where
        ``my_args`` and ``my_kwargs`` are required positional and keyword arguments.
        Rather than passing ``f0`` as the callable, wrap it to accept
        only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the
        callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been
        gathered before invoking this function.

    bracket : sequence, optional
        For methods 'brent' and 'golden', `bracket` defines the bracketing
        interval and is required.
        Either a triple ``(xa, xb, xc)`` satisfying ``xa < xb < xc`` and
        ``func(xb) < func(xa) and  func(xb) < func(xc)``, or a pair
        ``(xa, xb)`` to be used as initial points for a downhill bracket search
        (see `scipy.optimize.bracket`).
        The minimizer ``res.x`` will not necessarily satisfy
        ``xa <= res.x <= xb``.
    bounds : sequence, optional
        For method 'bounded', `bounds` is mandatory and must have two finite
        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:

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

        Default is "Bounded" if bounds are provided and "Brent" otherwise.
        See the 'Notes' section for details of each solver.

    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 the ``"Bounded"`` Brent method if
    `bounds` are passed and unbounded ``"Brent"`` otherwise.

    Method :ref:`Brent <optimize.minimize_scalar-brent>` uses Brent's
    algorithm [1]_ 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 [1]_. 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 [2]_ [3]_. It uses the Brent method to find a
    local minimum in the interval x1 < xopt < x2.

    Note that the Brent and Golden methods do not guarantee success unless a
    valid ``bracket`` triple is provided. If a three-point bracket cannot be
    found, consider `scipy.optimize.minimize`. Also, all methods are intended
    only for local minimization. When the function of interest has more than
    one local minimum, consider :ref:`global_optimization`.

    **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

    References
    ----------
    .. [1] Press, W., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery.
           Numerical Recipes in C. Cambridge University Press.
    .. [2] Forsythe, G.E., M. A. Malcolm, and C. B. Moler. "Computer Methods
           for Mathematical Computations." Prentice-Hall Series in Automatic
           Computation 259 (1977).
    .. [3] Brent, Richard P. Algorithms for Minimization Without Derivatives.
           Courier Corporation, 2013.

    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.fun
    -9.9149495908

    The minimizer is:

    >>> 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.fun  # minimum
    3.28365179850e-13
    >>> res.x  # minimizer
    -2.0000002026

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

    if callable(method):
        meth = "_custom"
    elif method is None:
        meth = 'brent' if bounds is None else 'bounded'
    else:
        meth = method.lower()
    if options is None:
        options = {}

    if bounds is not None and meth in {'brent', 'golden'}:
        message = f"Use of `bounds` is incompatible with 'method={method}'."
        raise ValueError(message)

    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, stacklevel=2)
            options['xatol'] = tol
        elif meth == '_custom':
            options.setdefault('tol', tol)
        else:
            options.setdefault('xtol', tol)

    # replace boolean "disp" option, if specified, by an integer value.
    disp = options.get('disp')
    if isinstance(disp, bool):
        options['disp'] = 2 * int(disp)

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

    # gh-16196 reported inconsistencies in the output shape of `res.x`. While
    # fixing this, future-proof it for when the function is vectorized:
    # the shape of `res.x` should match that of `res.fun`.
    res.fun = np.asarray(res.fun)[()]
    res.x = np.reshape(res.x, res.fun.shape)[()]
    return res


def _remove_from_bounds(bounds, i_fixed):
    """Removes fixed variables from a `Bounds` instance"""
    lb = bounds.lb[~i_fixed]
    ub = bounds.ub[~i_fixed]
    return Bounds(lb, ub)  # don't mutate original Bounds object


class _Patch_Callback_Equal_Variables:
    # Patches a callback that accepts an intermediate_result
    def __init__(self, callback, i_fixed, x_fixed):
        self.callback = callback
        self.i_fixed = i_fixed
        self.x_fixed = x_fixed

    def __call__(self, intermediate_result):
        x_in = intermediate_result.x
        x_out = np.zeros_like(self.i_fixed, dtype=x_in.dtype)
        x_out[self.i_fixed] = self.x_fixed
        x_out[~self.i_fixed] = x_in
        intermediate_result.x = x_out
        return self.callback(intermediate_result)


class _Remove_From_Func:
    """Wraps a function such that fixed variables need not be passed in"""
    def __init__(self, fun_in, i_fixed, x_fixed, min_dim=None, remove=0):
        self.fun_in = fun_in
        self.i_fixed = i_fixed
        self.x_fixed = x_fixed
        self.min_dim = min_dim
        self.remove = remove

    def __call__(self, x_in, *args, **kwargs):
        x_out = np.zeros_like(self.i_fixed, dtype=x_in.dtype)
        x_out[self.i_fixed] = self.x_fixed
        x_out[~self.i_fixed] = x_in
        y_out = self.fun_in(x_out, *args, **kwargs)
        y_out = np.array(y_out)

        if self.min_dim == 1:
            y_out = np.atleast_1d(y_out)
        elif self.min_dim == 2:
            y_out = np.atleast_2d(y_out)

        if self.remove == 1:
            y_out = y_out[..., ~self.i_fixed]
        elif self.remove == 2:
            y_out = y_out[~self.i_fixed, ~self.i_fixed]

        return y_out


def _add_to_array(x_in, i_fixed, x_fixed):
    """Adds fixed variables back to an array"""
    i_free = ~i_fixed
    if x_in.ndim == 2:
        i_free = i_free[:, None] @ i_free[None, :]
    x_out = np.zeros_like(i_free, dtype=x_in.dtype)
    x_out[~i_free] = x_fixed
    x_out[i_free] = x_in.ravel()
    return x_out


def _validate_bounds(bounds, x0, meth):
    """Check that bounds are valid."""

    msg = "An upper bound is less than the corresponding lower bound."
    if np.any(bounds.ub < bounds.lb):
        raise ValueError(msg)

    msg = "The number of bounds is not compatible with the length of `x0`."
    try:
        bounds.lb = np.broadcast_to(bounds.lb, x0.shape)
        bounds.ub = np.broadcast_to(bounds.ub, x0.shape)
    except Exception as e:
        raise ValueError(msg) from e

    return bounds

def standardize_bounds(bounds, x0, meth):
    """Converts bounds to the form required by the solver."""
    if meth in {'trust-constr', 'powell', 'nelder-mead', 'cobyla', 'cobyqa',
                'new'}:
        if not isinstance(bounds, Bounds):
            lb, ub = old_bound_to_new(bounds)
            bounds = Bounds(lb, ub)
    elif meth in ('l-bfgs-b', 'tnc', 'slsqp', 'old'):
        if isinstance(bounds, Bounds):
            bounds = new_bounds_to_old(bounds.lb, bounds.ub, x0.shape[0])
    return bounds


def standardize_constraints(constraints, x0, meth):
    """Converts constraints to the form required by the solver."""
    all_constraint_types = (NonlinearConstraint, LinearConstraint, dict)
    new_constraint_types = all_constraint_types[:-1]
    if constraints is None:
        constraints = []
    elif isinstance(constraints, all_constraint_types):
        constraints = [constraints]
    else:
        constraints = list(constraints)  # ensure it's a mutable sequence

    if meth in ['trust-constr', 'cobyqa', 'new', 'cobyla']:
        for i, con in enumerate(constraints):
            if not isinstance(con, new_constraint_types):
                constraints[i] = old_constraint_to_new(i, con)
    else:
        # iterate over copy, changing original
        for i, con in enumerate(list(constraints)):
            if isinstance(con, new_constraint_types):
                old_constraints = new_constraint_to_old(con, x0)
                constraints[i] = old_constraints[0]
                constraints.extend(old_constraints[1:])  # appends 1 if present

    return constraints


def _optimize_result_for_equal_bounds(
        fun, bounds, method, args=(), constraints=()
):
    """
    Provides a default OptimizeResult for when a bounded minimization method
    has (lb == ub).all().

    Parameters
    ----------
    fun: callable
    bounds: Bounds
    method: str
    constraints: Constraint
    """
    success = True
    message = 'All independent variables were fixed by bounds.'

    # bounds is new-style
    x0 = bounds.lb

    if constraints:
        message = ("All independent variables were fixed by bounds at values"
                   " that satisfy the constraints.")
        constraints = standardize_constraints(constraints, x0, 'new')

    maxcv = 0
    for c in constraints:
        pc = PreparedConstraint(c, x0)
        violation = pc.violation(x0)
        if np.sum(violation):
            maxcv = max(maxcv, np.max(violation))
            success = False
            message = (f"All independent variables were fixed by bounds, but "
                       f"the independent variables do not satisfy the "
                       f"constraints exactly. (Maximum violation: {maxcv}).")

    return OptimizeResult(
        x=x0, fun=fun(x0, *args), success=success, message=message, nfev=1,
        njev=0, nhev=0,
    )