897 lines
27 KiB
Python
897 lines
27 KiB
Python
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"""
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Functions
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---------
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.. autosummary::
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:toctree: generated/
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line_search_armijo
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line_search_wolfe1
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line_search_wolfe2
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scalar_search_wolfe1
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scalar_search_wolfe2
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"""
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from warnings import warn
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from ._dcsrch import DCSRCH
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import numpy as np
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__all__ = ['LineSearchWarning', 'line_search_wolfe1', 'line_search_wolfe2',
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'scalar_search_wolfe1', 'scalar_search_wolfe2',
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'line_search_armijo']
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class LineSearchWarning(RuntimeWarning):
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pass
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def _check_c1_c2(c1, c2):
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if not (0 < c1 < c2 < 1):
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raise ValueError("'c1' and 'c2' do not satisfy"
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"'0 < c1 < c2 < 1'.")
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#------------------------------------------------------------------------------
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# Minpack's Wolfe line and scalar searches
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#------------------------------------------------------------------------------
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def line_search_wolfe1(f, fprime, xk, pk, gfk=None,
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old_fval=None, old_old_fval=None,
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args=(), c1=1e-4, c2=0.9, amax=50, amin=1e-8,
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xtol=1e-14):
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"""
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As `scalar_search_wolfe1` but do a line search to direction `pk`
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Parameters
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----------
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f : callable
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Function `f(x)`
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fprime : callable
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Gradient of `f`
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xk : array_like
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Current point
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pk : array_like
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Search direction
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gfk : array_like, optional
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Gradient of `f` at point `xk`
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old_fval : float, optional
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Value of `f` at point `xk`
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old_old_fval : float, optional
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Value of `f` at point preceding `xk`
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The rest of the parameters are the same as for `scalar_search_wolfe1`.
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Returns
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-------
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stp, f_count, g_count, fval, old_fval
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As in `line_search_wolfe1`
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gval : array
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Gradient of `f` at the final point
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Notes
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-----
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Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1``.
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"""
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if gfk is None:
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gfk = fprime(xk, *args)
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gval = [gfk]
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gc = [0]
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fc = [0]
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def phi(s):
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fc[0] += 1
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return f(xk + s*pk, *args)
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def derphi(s):
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gval[0] = fprime(xk + s*pk, *args)
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gc[0] += 1
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return np.dot(gval[0], pk)
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derphi0 = np.dot(gfk, pk)
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stp, fval, old_fval = scalar_search_wolfe1(
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phi, derphi, old_fval, old_old_fval, derphi0,
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c1=c1, c2=c2, amax=amax, amin=amin, xtol=xtol)
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return stp, fc[0], gc[0], fval, old_fval, gval[0]
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def scalar_search_wolfe1(phi, derphi, phi0=None, old_phi0=None, derphi0=None,
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c1=1e-4, c2=0.9,
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amax=50, amin=1e-8, xtol=1e-14):
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"""
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Scalar function search for alpha that satisfies strong Wolfe conditions
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alpha > 0 is assumed to be a descent direction.
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Parameters
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----------
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phi : callable phi(alpha)
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Function at point `alpha`
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derphi : callable phi'(alpha)
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Objective function derivative. Returns a scalar.
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phi0 : float, optional
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Value of phi at 0
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old_phi0 : float, optional
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Value of phi at previous point
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derphi0 : float, optional
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Value derphi at 0
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c1 : float, optional
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Parameter for Armijo condition rule.
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c2 : float, optional
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Parameter for curvature condition rule.
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amax, amin : float, optional
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Maximum and minimum step size
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xtol : float, optional
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Relative tolerance for an acceptable step.
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Returns
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-------
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alpha : float
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Step size, or None if no suitable step was found
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phi : float
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Value of `phi` at the new point `alpha`
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phi0 : float
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Value of `phi` at `alpha=0`
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Notes
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-----
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Uses routine DCSRCH from MINPACK.
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Parameters `c1` and `c2` must satisfy ``0 < c1 < c2 < 1`` as described in [1]_.
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References
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----------
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.. [1] Nocedal, J., & Wright, S. J. (2006). Numerical optimization.
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In Springer Series in Operations Research and Financial Engineering.
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(Springer Series in Operations Research and Financial Engineering).
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Springer Nature.
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"""
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_check_c1_c2(c1, c2)
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if phi0 is None:
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phi0 = phi(0.)
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if derphi0 is None:
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derphi0 = derphi(0.)
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if old_phi0 is not None and derphi0 != 0:
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alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
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if alpha1 < 0:
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alpha1 = 1.0
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else:
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alpha1 = 1.0
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maxiter = 100
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dcsrch = DCSRCH(phi, derphi, c1, c2, xtol, amin, amax)
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stp, phi1, phi0, task = dcsrch(
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alpha1, phi0=phi0, derphi0=derphi0, maxiter=maxiter
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)
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return stp, phi1, phi0
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line_search = line_search_wolfe1
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#------------------------------------------------------------------------------
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# Pure-Python Wolfe line and scalar searches
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#------------------------------------------------------------------------------
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# Note: `line_search_wolfe2` is the public `scipy.optimize.line_search`
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def line_search_wolfe2(f, myfprime, xk, pk, gfk=None, old_fval=None,
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old_old_fval=None, args=(), c1=1e-4, c2=0.9, amax=None,
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extra_condition=None, maxiter=10):
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"""Find alpha that satisfies strong Wolfe conditions.
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Parameters
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----------
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f : callable f(x,*args)
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Objective function.
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myfprime : callable f'(x,*args)
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Objective function gradient.
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xk : ndarray
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Starting point.
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pk : ndarray
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Search direction. The search direction must be a descent direction
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for the algorithm to converge.
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gfk : ndarray, optional
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Gradient value for x=xk (xk being the current parameter
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estimate). Will be recomputed if omitted.
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old_fval : float, optional
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Function value for x=xk. Will be recomputed if omitted.
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old_old_fval : float, optional
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Function value for the point preceding x=xk.
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args : tuple, optional
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Additional arguments passed to objective function.
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c1 : float, optional
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Parameter for Armijo condition rule.
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c2 : float, optional
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Parameter for curvature condition rule.
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amax : float, optional
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Maximum step size
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extra_condition : callable, optional
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A callable of the form ``extra_condition(alpha, x, f, g)``
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returning a boolean. Arguments are the proposed step ``alpha``
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and the corresponding ``x``, ``f`` and ``g`` values. The line search
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accepts the value of ``alpha`` only if this
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callable returns ``True``. If the callable returns ``False``
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for the step length, the algorithm will continue with
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new iterates. The callable is only called for iterates
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satisfying the strong Wolfe conditions.
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maxiter : int, optional
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Maximum number of iterations to perform.
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Returns
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-------
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alpha : float or None
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Alpha for which ``x_new = x0 + alpha * pk``,
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or None if the line search algorithm did not converge.
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fc : int
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Number of function evaluations made.
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gc : int
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Number of gradient evaluations made.
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new_fval : float or None
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New function value ``f(x_new)=f(x0+alpha*pk)``,
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or None if the line search algorithm did not converge.
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old_fval : float
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Old function value ``f(x0)``.
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new_slope : float or None
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The local slope along the search direction at the
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new value ``<myfprime(x_new), pk>``,
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or None if the line search algorithm did not converge.
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Notes
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-----
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Uses the line search algorithm to enforce strong Wolfe
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conditions. See Wright and Nocedal, 'Numerical Optimization',
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1999, pp. 59-61.
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The search direction `pk` must be a descent direction (e.g.
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``-myfprime(xk)``) to find a step length that satisfies the strong Wolfe
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conditions. If the search direction is not a descent direction (e.g.
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``myfprime(xk)``), then `alpha`, `new_fval`, and `new_slope` will be None.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.optimize import line_search
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A objective function and its gradient are defined.
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>>> def obj_func(x):
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... return (x[0])**2+(x[1])**2
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>>> def obj_grad(x):
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... return [2*x[0], 2*x[1]]
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We can find alpha that satisfies strong Wolfe conditions.
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>>> start_point = np.array([1.8, 1.7])
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>>> search_gradient = np.array([-1.0, -1.0])
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>>> line_search(obj_func, obj_grad, start_point, search_gradient)
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(1.0, 2, 1, 1.1300000000000001, 6.13, [1.6, 1.4])
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"""
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fc = [0]
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gc = [0]
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gval = [None]
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gval_alpha = [None]
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def phi(alpha):
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fc[0] += 1
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return f(xk + alpha * pk, *args)
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fprime = myfprime
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def derphi(alpha):
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gc[0] += 1
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gval[0] = fprime(xk + alpha * pk, *args) # store for later use
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gval_alpha[0] = alpha
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return np.dot(gval[0], pk)
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if gfk is None:
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gfk = fprime(xk, *args)
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derphi0 = np.dot(gfk, pk)
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if extra_condition is not None:
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# Add the current gradient as argument, to avoid needless
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# re-evaluation
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def extra_condition2(alpha, phi):
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if gval_alpha[0] != alpha:
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derphi(alpha)
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x = xk + alpha * pk
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return extra_condition(alpha, x, phi, gval[0])
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else:
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extra_condition2 = None
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alpha_star, phi_star, old_fval, derphi_star = scalar_search_wolfe2(
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phi, derphi, old_fval, old_old_fval, derphi0, c1, c2, amax,
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extra_condition2, maxiter=maxiter)
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if derphi_star is None:
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warn('The line search algorithm did not converge',
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LineSearchWarning, stacklevel=2)
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else:
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# derphi_star is a number (derphi) -- so use the most recently
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# calculated gradient used in computing it derphi = gfk*pk
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# this is the gradient at the next step no need to compute it
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# again in the outer loop.
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derphi_star = gval[0]
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return alpha_star, fc[0], gc[0], phi_star, old_fval, derphi_star
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def scalar_search_wolfe2(phi, derphi, phi0=None,
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old_phi0=None, derphi0=None,
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c1=1e-4, c2=0.9, amax=None,
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extra_condition=None, maxiter=10):
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"""Find alpha that satisfies strong Wolfe conditions.
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alpha > 0 is assumed to be a descent direction.
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Parameters
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----------
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phi : callable phi(alpha)
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Objective scalar function.
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derphi : callable phi'(alpha)
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Objective function derivative. Returns a scalar.
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phi0 : float, optional
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Value of phi at 0.
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old_phi0 : float, optional
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Value of phi at previous point.
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derphi0 : float, optional
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Value of derphi at 0
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c1 : float, optional
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Parameter for Armijo condition rule.
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c2 : float, optional
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Parameter for curvature condition rule.
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amax : float, optional
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Maximum step size.
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extra_condition : callable, optional
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A callable of the form ``extra_condition(alpha, phi_value)``
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returning a boolean. The line search accepts the value
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of ``alpha`` only if this callable returns ``True``.
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If the callable returns ``False`` for the step length,
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the algorithm will continue with new iterates.
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The callable is only called for iterates satisfying
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the strong Wolfe conditions.
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maxiter : int, optional
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Maximum number of iterations to perform.
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Returns
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-------
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alpha_star : float or None
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Best alpha, or None if the line search algorithm did not converge.
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phi_star : float
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phi at alpha_star.
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phi0 : float
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phi at 0.
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derphi_star : float or None
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derphi at alpha_star, or None if the line search algorithm
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did not converge.
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Notes
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-----
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|
|
Uses the line search algorithm to enforce strong Wolfe
|
||
|
|
conditions. See Wright and Nocedal, 'Numerical Optimization',
|
||
|
|
1999, pp. 59-61.
|
||
|
|
|
||
|
|
"""
|
||
|
|
_check_c1_c2(c1, c2)
|
||
|
|
|
||
|
|
if phi0 is None:
|
||
|
|
phi0 = phi(0.)
|
||
|
|
|
||
|
|
if derphi0 is None:
|
||
|
|
derphi0 = derphi(0.)
|
||
|
|
|
||
|
|
alpha0 = 0
|
||
|
|
if old_phi0 is not None and derphi0 != 0:
|
||
|
|
alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
|
||
|
|
else:
|
||
|
|
alpha1 = 1.0
|
||
|
|
|
||
|
|
if alpha1 < 0:
|
||
|
|
alpha1 = 1.0
|
||
|
|
|
||
|
|
if amax is not None:
|
||
|
|
alpha1 = min(alpha1, amax)
|
||
|
|
|
||
|
|
phi_a1 = phi(alpha1)
|
||
|
|
#derphi_a1 = derphi(alpha1) evaluated below
|
||
|
|
|
||
|
|
phi_a0 = phi0
|
||
|
|
derphi_a0 = derphi0
|
||
|
|
|
||
|
|
if extra_condition is None:
|
||
|
|
def extra_condition(alpha, phi):
|
||
|
|
return True
|
||
|
|
|
||
|
|
for i in range(maxiter):
|
||
|
|
if alpha1 == 0 or (amax is not None and alpha0 > amax):
|
||
|
|
# alpha1 == 0: This shouldn't happen. Perhaps the increment has
|
||
|
|
# slipped below machine precision?
|
||
|
|
alpha_star = None
|
||
|
|
phi_star = phi0
|
||
|
|
phi0 = old_phi0
|
||
|
|
derphi_star = None
|
||
|
|
|
||
|
|
if alpha1 == 0:
|
||
|
|
msg = 'Rounding errors prevent the line search from converging'
|
||
|
|
else:
|
||
|
|
msg = "The line search algorithm could not find a solution " + \
|
||
|
|
"less than or equal to amax: %s" % amax
|
||
|
|
|
||
|
|
warn(msg, LineSearchWarning, stacklevel=2)
|
||
|
|
break
|
||
|
|
|
||
|
|
not_first_iteration = i > 0
|
||
|
|
if (phi_a1 > phi0 + c1 * alpha1 * derphi0) or \
|
||
|
|
((phi_a1 >= phi_a0) and not_first_iteration):
|
||
|
|
alpha_star, phi_star, derphi_star = \
|
||
|
|
_zoom(alpha0, alpha1, phi_a0,
|
||
|
|
phi_a1, derphi_a0, phi, derphi,
|
||
|
|
phi0, derphi0, c1, c2, extra_condition)
|
||
|
|
break
|
||
|
|
|
||
|
|
derphi_a1 = derphi(alpha1)
|
||
|
|
if (abs(derphi_a1) <= -c2*derphi0):
|
||
|
|
if extra_condition(alpha1, phi_a1):
|
||
|
|
alpha_star = alpha1
|
||
|
|
phi_star = phi_a1
|
||
|
|
derphi_star = derphi_a1
|
||
|
|
break
|
||
|
|
|
||
|
|
if (derphi_a1 >= 0):
|
||
|
|
alpha_star, phi_star, derphi_star = \
|
||
|
|
_zoom(alpha1, alpha0, phi_a1,
|
||
|
|
phi_a0, derphi_a1, phi, derphi,
|
||
|
|
phi0, derphi0, c1, c2, extra_condition)
|
||
|
|
break
|
||
|
|
|
||
|
|
alpha2 = 2 * alpha1 # increase by factor of two on each iteration
|
||
|
|
if amax is not None:
|
||
|
|
alpha2 = min(alpha2, amax)
|
||
|
|
alpha0 = alpha1
|
||
|
|
alpha1 = alpha2
|
||
|
|
phi_a0 = phi_a1
|
||
|
|
phi_a1 = phi(alpha1)
|
||
|
|
derphi_a0 = derphi_a1
|
||
|
|
|
||
|
|
else:
|
||
|
|
# stopping test maxiter reached
|
||
|
|
alpha_star = alpha1
|
||
|
|
phi_star = phi_a1
|
||
|
|
derphi_star = None
|
||
|
|
warn('The line search algorithm did not converge',
|
||
|
|
LineSearchWarning, stacklevel=2)
|
||
|
|
|
||
|
|
return alpha_star, phi_star, phi0, derphi_star
|
||
|
|
|
||
|
|
|
||
|
|
def _cubicmin(a, fa, fpa, b, fb, c, fc):
|
||
|
|
"""
|
||
|
|
Finds the minimizer for a cubic polynomial that goes through the
|
||
|
|
points (a,fa), (b,fb), and (c,fc) with derivative at a of fpa.
|
||
|
|
|
||
|
|
If no minimizer can be found, return None.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# f(x) = A *(x-a)^3 + B*(x-a)^2 + C*(x-a) + D
|
||
|
|
|
||
|
|
with np.errstate(divide='raise', over='raise', invalid='raise'):
|
||
|
|
try:
|
||
|
|
C = fpa
|
||
|
|
db = b - a
|
||
|
|
dc = c - a
|
||
|
|
denom = (db * dc) ** 2 * (db - dc)
|
||
|
|
d1 = np.empty((2, 2))
|
||
|
|
d1[0, 0] = dc ** 2
|
||
|
|
d1[0, 1] = -db ** 2
|
||
|
|
d1[1, 0] = -dc ** 3
|
||
|
|
d1[1, 1] = db ** 3
|
||
|
|
[A, B] = np.dot(d1, np.asarray([fb - fa - C * db,
|
||
|
|
fc - fa - C * dc]).flatten())
|
||
|
|
A /= denom
|
||
|
|
B /= denom
|
||
|
|
radical = B * B - 3 * A * C
|
||
|
|
xmin = a + (-B + np.sqrt(radical)) / (3 * A)
|
||
|
|
except ArithmeticError:
|
||
|
|
return None
|
||
|
|
if not np.isfinite(xmin):
|
||
|
|
return None
|
||
|
|
return xmin
|
||
|
|
|
||
|
|
|
||
|
|
def _quadmin(a, fa, fpa, b, fb):
|
||
|
|
"""
|
||
|
|
Finds the minimizer for a quadratic polynomial that goes through
|
||
|
|
the points (a,fa), (b,fb) with derivative at a of fpa.
|
||
|
|
|
||
|
|
"""
|
||
|
|
# f(x) = B*(x-a)^2 + C*(x-a) + D
|
||
|
|
with np.errstate(divide='raise', over='raise', invalid='raise'):
|
||
|
|
try:
|
||
|
|
D = fa
|
||
|
|
C = fpa
|
||
|
|
db = b - a * 1.0
|
||
|
|
B = (fb - D - C * db) / (db * db)
|
||
|
|
xmin = a - C / (2.0 * B)
|
||
|
|
except ArithmeticError:
|
||
|
|
return None
|
||
|
|
if not np.isfinite(xmin):
|
||
|
|
return None
|
||
|
|
return xmin
|
||
|
|
|
||
|
|
|
||
|
|
def _zoom(a_lo, a_hi, phi_lo, phi_hi, derphi_lo,
|
||
|
|
phi, derphi, phi0, derphi0, c1, c2, extra_condition):
|
||
|
|
"""Zoom stage of approximate linesearch satisfying strong Wolfe conditions.
|
||
|
|
|
||
|
|
Part of the optimization algorithm in `scalar_search_wolfe2`.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Implements Algorithm 3.6 (zoom) in Wright and Nocedal,
|
||
|
|
'Numerical Optimization', 1999, pp. 61.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
maxiter = 10
|
||
|
|
i = 0
|
||
|
|
delta1 = 0.2 # cubic interpolant check
|
||
|
|
delta2 = 0.1 # quadratic interpolant check
|
||
|
|
phi_rec = phi0
|
||
|
|
a_rec = 0
|
||
|
|
while True:
|
||
|
|
# interpolate to find a trial step length between a_lo and
|
||
|
|
# a_hi Need to choose interpolation here. Use cubic
|
||
|
|
# interpolation and then if the result is within delta *
|
||
|
|
# dalpha or outside of the interval bounded by a_lo or a_hi
|
||
|
|
# then use quadratic interpolation, if the result is still too
|
||
|
|
# close, then use bisection
|
||
|
|
|
||
|
|
dalpha = a_hi - a_lo
|
||
|
|
if dalpha < 0:
|
||
|
|
a, b = a_hi, a_lo
|
||
|
|
else:
|
||
|
|
a, b = a_lo, a_hi
|
||
|
|
|
||
|
|
# minimizer of cubic interpolant
|
||
|
|
# (uses phi_lo, derphi_lo, phi_hi, and the most recent value of phi)
|
||
|
|
#
|
||
|
|
# if the result is too close to the end points (or out of the
|
||
|
|
# interval), then use quadratic interpolation with phi_lo,
|
||
|
|
# derphi_lo and phi_hi if the result is still too close to the
|
||
|
|
# end points (or out of the interval) then use bisection
|
||
|
|
|
||
|
|
if (i > 0):
|
||
|
|
cchk = delta1 * dalpha
|
||
|
|
a_j = _cubicmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi,
|
||
|
|
a_rec, phi_rec)
|
||
|
|
if (i == 0) or (a_j is None) or (a_j > b - cchk) or (a_j < a + cchk):
|
||
|
|
qchk = delta2 * dalpha
|
||
|
|
a_j = _quadmin(a_lo, phi_lo, derphi_lo, a_hi, phi_hi)
|
||
|
|
if (a_j is None) or (a_j > b-qchk) or (a_j < a+qchk):
|
||
|
|
a_j = a_lo + 0.5*dalpha
|
||
|
|
|
||
|
|
# Check new value of a_j
|
||
|
|
|
||
|
|
phi_aj = phi(a_j)
|
||
|
|
if (phi_aj > phi0 + c1*a_j*derphi0) or (phi_aj >= phi_lo):
|
||
|
|
phi_rec = phi_hi
|
||
|
|
a_rec = a_hi
|
||
|
|
a_hi = a_j
|
||
|
|
phi_hi = phi_aj
|
||
|
|
else:
|
||
|
|
derphi_aj = derphi(a_j)
|
||
|
|
if abs(derphi_aj) <= -c2*derphi0 and extra_condition(a_j, phi_aj):
|
||
|
|
a_star = a_j
|
||
|
|
val_star = phi_aj
|
||
|
|
valprime_star = derphi_aj
|
||
|
|
break
|
||
|
|
if derphi_aj*(a_hi - a_lo) >= 0:
|
||
|
|
phi_rec = phi_hi
|
||
|
|
a_rec = a_hi
|
||
|
|
a_hi = a_lo
|
||
|
|
phi_hi = phi_lo
|
||
|
|
else:
|
||
|
|
phi_rec = phi_lo
|
||
|
|
a_rec = a_lo
|
||
|
|
a_lo = a_j
|
||
|
|
phi_lo = phi_aj
|
||
|
|
derphi_lo = derphi_aj
|
||
|
|
i += 1
|
||
|
|
if (i > maxiter):
|
||
|
|
# Failed to find a conforming step size
|
||
|
|
a_star = None
|
||
|
|
val_star = None
|
||
|
|
valprime_star = None
|
||
|
|
break
|
||
|
|
return a_star, val_star, valprime_star
|
||
|
|
|
||
|
|
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
# Armijo line and scalar searches
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
|
||
|
|
def line_search_armijo(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1):
|
||
|
|
"""Minimize over alpha, the function ``f(xk+alpha pk)``.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
f : callable
|
||
|
|
Function to be minimized.
|
||
|
|
xk : array_like
|
||
|
|
Current point.
|
||
|
|
pk : array_like
|
||
|
|
Search direction.
|
||
|
|
gfk : array_like
|
||
|
|
Gradient of `f` at point `xk`.
|
||
|
|
old_fval : float
|
||
|
|
Value of `f` at point `xk`.
|
||
|
|
args : tuple, optional
|
||
|
|
Optional arguments.
|
||
|
|
c1 : float, optional
|
||
|
|
Value to control stopping criterion.
|
||
|
|
alpha0 : scalar, optional
|
||
|
|
Value of `alpha` at start of the optimization.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
alpha
|
||
|
|
f_count
|
||
|
|
f_val_at_alpha
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Uses the interpolation algorithm (Armijo backtracking) as suggested by
|
||
|
|
Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57
|
||
|
|
|
||
|
|
"""
|
||
|
|
xk = np.atleast_1d(xk)
|
||
|
|
fc = [0]
|
||
|
|
|
||
|
|
def phi(alpha1):
|
||
|
|
fc[0] += 1
|
||
|
|
return f(xk + alpha1*pk, *args)
|
||
|
|
|
||
|
|
if old_fval is None:
|
||
|
|
phi0 = phi(0.)
|
||
|
|
else:
|
||
|
|
phi0 = old_fval # compute f(xk) -- done in past loop
|
||
|
|
|
||
|
|
derphi0 = np.dot(gfk, pk)
|
||
|
|
alpha, phi1 = scalar_search_armijo(phi, phi0, derphi0, c1=c1,
|
||
|
|
alpha0=alpha0)
|
||
|
|
return alpha, fc[0], phi1
|
||
|
|
|
||
|
|
|
||
|
|
def line_search_BFGS(f, xk, pk, gfk, old_fval, args=(), c1=1e-4, alpha0=1):
|
||
|
|
"""
|
||
|
|
Compatibility wrapper for `line_search_armijo`
|
||
|
|
"""
|
||
|
|
r = line_search_armijo(f, xk, pk, gfk, old_fval, args=args, c1=c1,
|
||
|
|
alpha0=alpha0)
|
||
|
|
return r[0], r[1], 0, r[2]
|
||
|
|
|
||
|
|
|
||
|
|
def scalar_search_armijo(phi, phi0, derphi0, c1=1e-4, alpha0=1, amin=0):
|
||
|
|
"""Minimize over alpha, the function ``phi(alpha)``.
|
||
|
|
|
||
|
|
Uses the interpolation algorithm (Armijo backtracking) as suggested by
|
||
|
|
Wright and Nocedal in 'Numerical Optimization', 1999, pp. 56-57
|
||
|
|
|
||
|
|
alpha > 0 is assumed to be a descent direction.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
alpha
|
||
|
|
phi1
|
||
|
|
|
||
|
|
"""
|
||
|
|
phi_a0 = phi(alpha0)
|
||
|
|
if phi_a0 <= phi0 + c1*alpha0*derphi0:
|
||
|
|
return alpha0, phi_a0
|
||
|
|
|
||
|
|
# Otherwise, compute the minimizer of a quadratic interpolant:
|
||
|
|
|
||
|
|
alpha1 = -(derphi0) * alpha0**2 / 2.0 / (phi_a0 - phi0 - derphi0 * alpha0)
|
||
|
|
phi_a1 = phi(alpha1)
|
||
|
|
|
||
|
|
if (phi_a1 <= phi0 + c1*alpha1*derphi0):
|
||
|
|
return alpha1, phi_a1
|
||
|
|
|
||
|
|
# Otherwise, loop with cubic interpolation until we find an alpha which
|
||
|
|
# satisfies the first Wolfe condition (since we are backtracking, we will
|
||
|
|
# assume that the value of alpha is not too small and satisfies the second
|
||
|
|
# condition.
|
||
|
|
|
||
|
|
while alpha1 > amin: # we are assuming alpha>0 is a descent direction
|
||
|
|
factor = alpha0**2 * alpha1**2 * (alpha1-alpha0)
|
||
|
|
a = alpha0**2 * (phi_a1 - phi0 - derphi0*alpha1) - \
|
||
|
|
alpha1**2 * (phi_a0 - phi0 - derphi0*alpha0)
|
||
|
|
a = a / factor
|
||
|
|
b = -alpha0**3 * (phi_a1 - phi0 - derphi0*alpha1) + \
|
||
|
|
alpha1**3 * (phi_a0 - phi0 - derphi0*alpha0)
|
||
|
|
b = b / factor
|
||
|
|
|
||
|
|
alpha2 = (-b + np.sqrt(abs(b**2 - 3 * a * derphi0))) / (3.0*a)
|
||
|
|
phi_a2 = phi(alpha2)
|
||
|
|
|
||
|
|
if (phi_a2 <= phi0 + c1*alpha2*derphi0):
|
||
|
|
return alpha2, phi_a2
|
||
|
|
|
||
|
|
if (alpha1 - alpha2) > alpha1 / 2.0 or (1 - alpha2/alpha1) < 0.96:
|
||
|
|
alpha2 = alpha1 / 2.0
|
||
|
|
|
||
|
|
alpha0 = alpha1
|
||
|
|
alpha1 = alpha2
|
||
|
|
phi_a0 = phi_a1
|
||
|
|
phi_a1 = phi_a2
|
||
|
|
|
||
|
|
# Failed to find a suitable step length
|
||
|
|
return None, phi_a1
|
||
|
|
|
||
|
|
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
# Non-monotone line search for DF-SANE
|
||
|
|
#------------------------------------------------------------------------------
|
||
|
|
|
||
|
|
def _nonmonotone_line_search_cruz(f, x_k, d, prev_fs, eta,
|
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gamma=1e-4, tau_min=0.1, tau_max=0.5):
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"""
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Nonmonotone backtracking line search as described in [1]_
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Parameters
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----------
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f : callable
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Function returning a tuple ``(f, F)`` where ``f`` is the value
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of a merit function and ``F`` the residual.
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x_k : ndarray
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Initial position.
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d : ndarray
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Search direction.
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prev_fs : float
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List of previous merit function values. Should have ``len(prev_fs) <= M``
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where ``M`` is the nonmonotonicity window parameter.
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eta : float
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Allowed merit function increase, see [1]_
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gamma, tau_min, tau_max : float, optional
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Search parameters, see [1]_
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Returns
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-------
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alpha : float
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Step length
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xp : ndarray
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Next position
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fp : float
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Merit function value at next position
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Fp : ndarray
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Residual at next position
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References
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----------
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[1] "Spectral residual method without gradient information for solving
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large-scale nonlinear systems of equations." W. La Cruz,
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J.M. Martinez, M. Raydan. Math. Comp. **75**, 1429 (2006).
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"""
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f_k = prev_fs[-1]
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f_bar = max(prev_fs)
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alpha_p = 1
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alpha_m = 1
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alpha = 1
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while True:
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xp = x_k + alpha_p * d
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fp, Fp = f(xp)
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if fp <= f_bar + eta - gamma * alpha_p**2 * f_k:
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alpha = alpha_p
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break
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alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k)
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xp = x_k - alpha_m * d
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fp, Fp = f(xp)
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if fp <= f_bar + eta - gamma * alpha_m**2 * f_k:
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alpha = -alpha_m
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break
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alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k)
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alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p)
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alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m)
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return alpha, xp, fp, Fp
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def _nonmonotone_line_search_cheng(f, x_k, d, f_k, C, Q, eta,
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gamma=1e-4, tau_min=0.1, tau_max=0.5,
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nu=0.85):
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"""
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Nonmonotone line search from [1]
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Parameters
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----------
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f : callable
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Function returning a tuple ``(f, F)`` where ``f`` is the value
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of a merit function and ``F`` the residual.
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x_k : ndarray
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Initial position.
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d : ndarray
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Search direction.
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f_k : float
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Initial merit function value.
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C, Q : float
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Control parameters. On the first iteration, give values
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Q=1.0, C=f_k
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eta : float
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Allowed merit function increase, see [1]_
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nu, gamma, tau_min, tau_max : float, optional
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Search parameters, see [1]_
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Returns
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-------
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alpha : float
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Step length
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xp : ndarray
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Next position
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fp : float
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Merit function value at next position
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Fp : ndarray
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Residual at next position
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C : float
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New value for the control parameter C
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Q : float
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New value for the control parameter Q
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References
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----------
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.. [1] W. Cheng & D.-H. Li, ''A derivative-free nonmonotone line
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search and its application to the spectral residual
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method'', IMA J. Numer. Anal. 29, 814 (2009).
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"""
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alpha_p = 1
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alpha_m = 1
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alpha = 1
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while True:
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xp = x_k + alpha_p * d
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fp, Fp = f(xp)
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if fp <= C + eta - gamma * alpha_p**2 * f_k:
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alpha = alpha_p
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break
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alpha_tp = alpha_p**2 * f_k / (fp + (2*alpha_p - 1)*f_k)
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xp = x_k - alpha_m * d
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fp, Fp = f(xp)
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if fp <= C + eta - gamma * alpha_m**2 * f_k:
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alpha = -alpha_m
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break
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alpha_tm = alpha_m**2 * f_k / (fp + (2*alpha_m - 1)*f_k)
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alpha_p = np.clip(alpha_tp, tau_min * alpha_p, tau_max * alpha_p)
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alpha_m = np.clip(alpha_tm, tau_min * alpha_m, tau_max * alpha_m)
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# Update C and Q
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Q_next = nu * Q + 1
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C = (nu * Q * (C + eta) + fp) / Q_next
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Q = Q_next
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return alpha, xp, fp, Fp, C, Q
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