monotonicity_test.py

from .globalization import Globalization

import warnings


class MonotonicityTest(Globalization):
    """Restricted residual monotonicity test

    The restricted residual monotonicity test computes the largest
    step size `sigma` in {1, 1/2, 1/4, ..., sigma_min} such that

        residual_map(sigma) <= (1 - sigma/100 + 2e-4)*residual_map_old.

    If such a parameter `sigma` does not exist, then `sigma = sigma_min/2`
    is chosen and a warning is printed.

    Instead of using 1/4 as in eq. (3.32) in Deulfhard (2011), we use 1/100.

    The term 2e-4 tries to take into account for potentially inaccurate evaluations
    of the residual mapping, assumed to be accurate up to four significant figures.

    The default value of `sigma_min` is taken from Mannel and Rund (2020).

    References:
    -----------

    P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Ser. Comput.
    Math. 35, Springer, Berlin, 2011, https://doi.org/10.1007/978-3-642-23899-4

    F. Mannel and A. Rund, A hybrid semismooth quasi-Newton method for
    nonsmooth optimal control with PDEs, Optim. Eng., (2020),
    https://doi.org/10.1007/s11081-020-09523-w
    """


    def __init__(self):

        self.sigma_min = .5**10


    def globalize(self, residual_map, residual_map_old, gtol):

        sigma = 1.0
        sigma_min = self.sigma_min

        residual_map_new = residual_map(sigma)

        while residual_map_new > (1.0-sigma/100.0+2e-4)*residual_map_old and sigma >= sigma_min:
            sigma /= 2.0
            residual_map_new = residual_map(sigma)

        if sigma <= sigma_min:
            warnings.warn("Step size sigma = {} is smaller than minimum step size = {}.".format(sigma, sigma_min))

        return sigma, residual_map_new