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This module implements the Sequential Least Squares Programming optimization
algorithm (SLSQP), originally developed by Dieter Kraft.
See http://www.netlib.org/toms/733

Functions
---------
.. autosummary::
   :toctree: generated/

    approx_jacobian
    fmin_slsqp

approx_jacobian
fmin_slsqp    N)slsqp)zerosarraylinalgappendasfarrayconcatenatefinfosqrtvstackexpinfisfinite
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    Approximate the Jacobian matrix of a callable function.

    Parameters
    ----------
    x : array_like
        The state vector at which to compute the Jacobian matrix.
    func : callable f(x,*args)
        The vector-valued function.
    epsilon : float
        The perturbation used to determine the partial derivatives.
    args : sequence
        Additional arguments passed to func.

    Returns
    -------
    An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length
    of the outputs of `func`, and ``lenx`` is the number of elements in
    `x`.

    Notes
    -----
    The approximation is done using forward differences.

    z2-point)methodabs_stepargs)r   npZ
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    Minimize a function using Sequential Least Squares Programming

    Python interface function for the SLSQP Optimization subroutine
    originally implemented by Dieter Kraft.

    Parameters
    ----------
    func : callable f(x,*args)
        Objective function.  Must return a scalar.
    x0 : 1-D ndarray of float
        Initial guess for the independent variable(s).
    eqcons : list, optional
        A list of functions of length n such that
        eqcons[j](x,*args) == 0.0 in a successfully optimized
        problem.
    f_eqcons : callable f(x,*args), optional
        Returns a 1-D array in which each element must equal 0.0 in a
        successfully optimized problem. If f_eqcons is specified,
        eqcons is ignored.
    ieqcons : list, optional
        A list of functions of length n such that
        ieqcons[j](x,*args) >= 0.0 in a successfully optimized
        problem.
    f_ieqcons : callable f(x,*args), optional
        Returns a 1-D ndarray in which each element must be greater or
        equal to 0.0 in a successfully optimized problem. If
        f_ieqcons is specified, ieqcons is ignored.
    bounds : list, optional
        A list of tuples specifying the lower and upper bound
        for each independent variable [(xl0, xu0),(xl1, xu1),...]
        Infinite values will be interpreted as large floating values.
    fprime : callable `f(x,*args)`, optional
        A function that evaluates the partial derivatives of func.
    fprime_eqcons : callable `f(x,*args)`, optional
        A function of the form `f(x, *args)` that returns the m by n
        array of equality constraint normals. If not provided,
        the normals will be approximated. The array returned by
        fprime_eqcons should be sized as ( len(eqcons), len(x0) ).
    fprime_ieqcons : callable `f(x,*args)`, optional
        A function of the form `f(x, *args)` that returns the m by n
        array of inequality constraint normals. If not provided,
        the normals will be approximated. The array returned by
        fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ).
    args : sequence, optional
        Additional arguments passed to func and fprime.
    iter : int, optional
        The maximum number of iterations.
    acc : float, optional
        Requested accuracy.
    iprint : int, optional
        The verbosity of fmin_slsqp :

        * iprint <= 0 : Silent operation
        * iprint == 1 : Print summary upon completion (default)
        * iprint >= 2 : Print status of each iterate and summary
    disp : int, optional
        Overrides the iprint interface (preferred).
    full_output : bool, optional
        If False, return only the minimizer of func (default).
        Otherwise, output final objective function and summary
        information.
    epsilon : float, optional
        The step size for finite-difference derivative estimates.
    callback : callable, optional
        Called after each iteration, as ``callback(x)``, where ``x`` is the
        current parameter vector.

    Returns
    -------
    out : ndarray of float
        The final minimizer of func.
    fx : ndarray of float, if full_output is true
        The final value of the objective function.
    its : int, if full_output is true
        The number of iterations.
    imode : int, if full_output is true
        The exit mode from the optimizer (see below).
    smode : string, if full_output is true
        Message describing the exit mode from the optimizer.

    See also
    --------
    minimize: Interface to minimization algorithms for multivariate
        functions. See the 'SLSQP' `method` in particular.

    Notes
    -----
    Exit modes are defined as follows ::

        -1 : Gradient evaluation required (g & a)
         0 : Optimization terminated successfully
         1 : Function evaluation required (f & c)
         2 : More equality constraints than independent variables
         3 : More than 3*n iterations in LSQ subproblem
         4 : Inequality constraints incompatible
         5 : Singular matrix E in LSQ subproblem
         6 : Singular matrix C in LSQ subproblem
         7 : Rank-deficient equality constraint subproblem HFTI
         8 : Positive directional derivative for linesearch
         9 : Iteration limit reached

    Examples
    --------
    Examples are given :ref:`in the tutorial <tutorial-sqlsp>`.

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    Minimize a scalar function of one or more variables using Sequential
    Least Squares Programming (SLSQP).

    Options
    -------
    ftol : float
        Precision goal for the value of f in the stopping criterion.
    eps : float
        Step size used for numerical approximation of the Jacobian.
    disp : bool
        Set to True to print convergence messages. If False,
        `verbosity` is ignored and set to 0.
    maxiter : int
        Maximum number of iterations.
    finite_diff_rel_step : None or array_like, optional
        If `jac in ['2-point', '3-point', 'cs']` the relative step size to
        use for numerical approximation of `jac`. The absolute step
        size is computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``,
        possibly adjusted to fit into the bounds. For ``method='3-point'``
        the sign of `h` is ignored. If None (default) then step is selected
        automatically.
    r   r   Nr#   )r,   r2   r-   z"Constraint %d has no type defined.z/Constraints must be defined using a dictionary.z#Constraint's type must be a string.zUnknown constraint type '%s'.r.   z&Constraint %d has no function defined.r"   c                s    fdd}|S )Nc                s>   t | } dkr&t| |dS t| d |dS d S )N)z2-pointz3-pointcs)r   r   Zrel_stepr3   z2-point)r   r   r   r3   )r   r   )r   r   )r!   finite_diff_rel_stepr.   r"   
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
z3_minimize_slsqp.<locals>.cjac_factory.<locals>.cjacr#   )r.   rH   )r!   rF   r"   rG   )r.   r$   cjac_factory%  s    z%_minimize_slsqp.<locals>.cjac_factoryr   )r.   r"   r   z$Gradient evaluation required (g & a)z$Optimization terminated successfullyz$Function evaluation required (f & c)z4More equality constraints than independent variablesz*More than 3*n iterations in LSQ subproblemz#Inequality constraints incompatiblez#Singular matrix E in LSQ subproblemz#Singular matrix C in LSQ subproblemz2Rank-deficient equality constraint subproblem HFTIz.Positive directional derivative for linesearchzIteration limit reached)r   r                        	   c                s&   g | ]}t |d   f|d  qS )r.   r   )r   )r/   r0   )r   r#   r$   
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isinstancedict	enumeratelowerKeyError	TypeErrorAttributeError
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r9   c                sh   |d r$t  fdd|d D }ntd}|d rPt  fdd|d D }ntd}t ||f}|S )Nr,   c                s&   g | ]}t |d   f|d  qS )r.   r   )r   )r/   ry   )r   r#   r$   rS     s   z$_eval_constraint.<locals>.<listcomp>r   r2   c                s&   g | ]}t |d   f|d  qS )r.   r   )r   )r/   ry   )r   r#   r$   rS     s   )r
   r   )r   rC   Zc_eqZc_ieqr0   r#   )r   r$   ru     s    ru   c       
         s   |d r$t  fdd|d D }nt||f}|d rTt  fdd|d D }nt||f}|dkrvt||f}	nt ||f}	t|	t|dgfd}	|	S )Nr,   c                s"   g | ]}|d   f|d  qS )r"   r   r#   )r/   ry   )r   r#   r$   rS     s   z%_eval_con_normals.<locals>.<listcomp>r2   c                s"   g | ]}|d   f|d  qS )r"   r   r#   )r/   ry   )r   r#   r$   rS     s   r   r   )r   r   r
   )
r   rC   r   r   r~   r|   r}   Za_eqZa_ieqr   r#   )r   r$   rv     s    rv   __main__rM   rK   c             C   sd   t | d |d | d d  |d | d d   |d | d  | d   |d | d   |d   S )z Objective function r   rK   r   rL   rM   )r   )r   rr#   r#   r$   r.     s    
Nr.   g?g?c             C   s   t | d d | d  | gS )z Equality constraint r   rK   r   )r   )r   rY   r#   r#   r$   feqcon  s    r   c             C   s   t d| d  dggS )z! Jacobian of equality constraint rK   r   r   )r   )r   rY   r#   r#   r$   jeqcon  s    r   
   c             C   s   t | d | d  | gS )z Inequality constraint r   r   )r   )r   r0   r#   r#   r$   fieqcon  s    r   c             C   s   t ddggS )z# Jacobian of inequality constraint r   )r   )r   r0   r#   r#   r$   jieqcon  s    r   r,   )r   )r-   r.   r"   r   r2   )r   z Bounds constraints H   -z * fmin_slsqprJ   T)r3   r)   rA   z * _minimize_slsqpr3   r)   z% Equality and inequality constraints )r;   r=   r<   r>   r)   rA   r4   )r   )r   )r   )r   )6__doc____all__warningsnumpyr   Zscipy.optimize._slsqpr   r   r   r   r   r	   r
   r   r   r   r   r   r   r   optimizer   r   r   r   r   Z_numdiffr   _constraintsr   r   __docformat__rl   r*   Z_epsilonr   r   r9   ru   rv   __name__r.   Tr   r   r   r   r   rC   rt   centerr   frD   r#   r#   r#   r$   <module>   sd   <"  |


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