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# scipy.optimize.fmin_l_bfgs_b

scipy.optimize.fmin_l_bfgs_b(func, x0, fprime=None, args=(), approx_grad=0, bounds=None, m=10, factr=10000000.0, pgtol=1e-05, epsilon=1e-08, iprint=-1, maxfun=15000, maxiter=15000, disp=None, callback=None, maxls=20)

  Minimize a function func using the L-BFGS-B algorithm.

  Parameters:
    func callable f(x,*args)

      Function to minimize.

    x0 ndarray

      Initial guess.

    fprime callable fprime(x,*args), optional

      The gradient of func. If None, then func returns the function value and the gradient (`f, g = func(x, *args)`), unless approx_grad is True in which case func returns only `f`.

    args sequence, optional

      Arguments to pass to func and fprime.

    approx_grad bool, optional

      Whether to approximate the gradient numerically (in which case func returns only the function value).

    bounds list, optional

      `(min, max)` pairs for each element in `x`, defining the bounds on that parameter. Use None or +-inf for one of `min` or `max` when there is no bound in that direction.

    m int, optional

      The maximum number of variable metric corrections used to define the limited memory matrix. (The limited memory BFGS method does not store the full hessian but uses this many terms in an approximation to it.)

    factr float, optional

      The iteration stops when `(f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr * eps`, where `eps` is the machine precision, which is automatically generated by the code. Typical values for factr are: 1e12 for low accuracy; 1e7 for moderate accuracy; 10.0 for extremely high accuracy. See Notes for relationship to ftol, which is exposed (instead of factr) by the `scipy.optimize.minimize` interface to L-BFGS-B.

    pgtol float, optional

      The iteration will stop when `max{|proj g_i | i = 1, ..., n} <= pgtol` where `proj g_i` is the i-th component of the projected gradient.

    epsilon float, optional

      Step size used when approx_grad is True, for numerically calculating the gradient

    iprint int, optional

      Controls the frequency of output. `iprint < 0` means no output; `iprint = 0` print only one line at the last iteration; `0 < iprint < 99` print also f and `|proj g|` every iprint iterations; `iprint = 99` print details of every iteration except n-vectors; `iprint = 100` print also the changes of active set and final x; `iprint > 100` print details of every iteration including x and g.

    disp int, optional

      If zero, then no output. If a positive number, then this over-rides iprint (i.e., iprint gets the value of disp).

    maxfun int, optional

      Maximum number of function evaluations. Note that this function may violate the limit because of evaluating gradients by numerical differentiation.

    maxiter int, optional

      Maximum number of iterations.

    callback callable, optional

      Called after each iteration, as `callback(xk)`, where `xk` is the current parameter vector.

    maxls int, optional

      Maximum number of line search steps (per iteration). Default is 20.

  Returns:
    x array_like

      Estimated position of the minimum.

    f float

      Value of func at the minimum.

    d dict

      Information dictionary.

      - d[‘warnflag’] is

        - 0 if converged,

        - 1 if too many function evaluations or too many iterations,

        - 2 if stopped for another reason, given in d[‘task’]

      - d[‘grad’] is the gradient at the minimum (should be 0 ish)

      - d[‘funcalls’] is the number of function calls made.

      - d[‘nit’] is the number of iterations.

  See also

  `minimize`

    Interface to minimization algorithms for multivariate functions. See the ‘L-BFGS-B’ method in particular. Note that the ftol option is made available via that interface, while factr is provided via this interface, where factr is the factor multiplying the default machine floating-point precision to arrive at ftol: `ftol = factr * numpy.finfo(float).eps`.

  Notes

  License of L-BFGS-B (FORTRAN code):

  The version included here (in fortran code) is 3.0 (released April 25, 2011). It was written by Ciyou Zhu, Richard Byrd, and Jorge Nocedal <nocedal@ece.nwu.edu>. It carries the following condition for use:

  This software is freely available, but we expect that all publications describing work using this software, or all commercial products using it, quote at least one of the references given below. This software is released under the BSD License.

  References

  - R. H. Byrd, P. Lu and J. Nocedal. A Limited Memory Algorithm for Bound Constrained Optimization, (1995), SIAM Journal on Scientific and Statistical Computing, 16, 5, pp. 1190-1208.

  - C. Zhu, R. H. Byrd and J. Nocedal. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (1997), ACM Transactions on Mathematical Software, 23, 4, pp. 550 - 560.

  - J.L. Morales and J. Nocedal. L-BFGS-B: Remark on Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization (2011), ACM Transactions on Mathematical Software, 38, 1.

  Examples

  Solve a linear regression problem via `fmin_l_bfgs_b`. To do this, first we define an objective function `f(m, b) = (y - y_model)**2`, where y describes the observations and y_model the prediction of the linear model as `y_model = m*x + b`. The bounds for the parameters, `m` and `b`, are arbitrarily chosen as `(0,5)` and `(5,10)` for this example.

      >>> import numpy as np
      >>> from scipy.optimize import fmin_l_bfgs_b
      >>> X = np.arange(0, 10, 1)
      >>> M = 2
      >>> B = 3
      >>> Y = M * X + B
      >>> def func(parameters, *args):
      ...     x = args[0]
      ...     y = args[1]
      ...     m, b = parameters
      ...     y_model = m*x + b
      ...     error = sum(np.power((y - y_model), 2))
      ...     return error

      >>> initial_values = np.array([0.0, 1.0])

      >>> x_opt, f_opt, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y),
      ...                                    approx_grad=True)
      >>> x_opt, f_opt
      array([1.99999999, 3.00000006]), 1.7746231151323805e-14  # may vary

  The optimized parameters in `x_opt` agree with the ground truth parameters `m` and `b`. Next, let us perform a bound contrained optimization using the bounds parameter.

      >>> bounds = [(0, 5), (5, 10)]
      >>> x_opt, f_op, info = fmin_l_bfgs_b(func, x0=initial_values, args=(X, Y),
      ...                                   approx_grad=True, bounds=bounds)
      >>> x_opt, f_opt
      array([1.65990508, 5.31649385]), 15.721334516453945  # may vary