Source code for pyproximal.projection.Box

import numpy as np
from scipy.optimize import bisect

[docs]class BoxProj(): r"""Box orthogonal projection. Parameters ---------- lower : :obj:`float` or :obj:`np.ndarray`, optional Lower bound upper : :obj:`float` or :obj:`np.ndarray`, optional Upper bound Notes ----- Given a Box set defined as: .. math:: \operatorname{Box}_{[l, u]} = \{ x: l \leq x\leq u \} its orthogonal projection is: .. math:: P_{\operatorname{Box}_{[l, u]}} (x_i) = min\{ max \{x_i, l_i\}, u_i \} = \begin{cases} l_i, & x_i < l_i\\ x_i,& l_i \leq x_i \leq u_i \\ u_i, & x_i > u_i\\ \end{cases} \quad \forall i Note that this is the proximal operator of the corresponding indicator function :math:`\mathcal{I}_{\operatorname{Box}_{[l, u]}}`. """ def __init__(self, lower=-np.inf, upper=np.inf): self.lower = lower self.upper = upper def __call__(self, x): x = np.minimum(np.maximum(x, self.lower), self.upper) return x
[docs]class HyperPlaneBoxProj(): r"""Orthogonal projection of the intersection between a Hyperplane and a Box. Parameters ---------- coeffs : :obj:`np.ndarray` Vector of coefficients used in the definition of the hyperplane scalar : :obj:`float` Scalar used in the definition of the hyperplane lower : :obj:`float` or :obj:`np.ndarray`, optional Lower bound of Box upper : :obj:`float` or :obj:`np.ndarray`, optional Upper bound of Box maxiter : :obj:`int`, optional Maximum number of iterations used by :func:`scipy.optimize.bisect` xtol : :obj:`float`, optional Absolute tolerance of :func:`scipy.optimize.bisect` Notes ----- Given the definition of an Hyperplane: .. math:: H_{c,b} = \{ \mathbf{x}: \mathbf{c}^T \mathbf{x} = b\} that of a Box (see :class:`pyproximal.projection.Box.BoxProj`), the intersection between the two can be written as: .. math:: C = Box_{[l, u]} \cap H_{c,b} = \{ \mathbf{x}: \mathbf{c}^T \mathbf{x} = b , \; l \leq x_i \leq u \} The orthogonal projection of such intersection is given by: .. math:: P_C = P_{Box_{[l, u]}} (\mathbf{x} - \mu^* \mathbf{c}) where :math:`\mu` is obtained by solving the following equation by bisection .. math:: f(\mu) = \mathbf{c}^T P_{Box_{[l, u]}} (\mathbf{x} - \mu \mathbf{c}) - b """ def __init__(self, coeffs, scalar, lower=-np.inf, upper=np.inf, maxiter=100, xtol=1e-5): self.coeffs = coeffs.ravel() self.scalar = scalar self.lower = lower self.upper = upper self.maxiter = maxiter self.xtol = xtol = BoxProj(lower, upper) def __call__(self, x): """Apply HyperPlaneBoxProj projection Parameters ---------- x : :obj:`np.ndarray` Vector """ def fun(mu, x): return, - mu * self.coeffs)) - \ self.scalar xshape = x.shape x = x.ravel() # identify brackets for bisect ensuring that the evaluated fun # has different sign bisect_lower = -1 while fun(bisect_lower, x) < 0: bisect_lower *= 2 bisect_upper = 1 while fun(bisect_upper, x) > 0: bisect_upper *= 2 # find optimal mu mu = bisect(lambda mu: fun(mu, x), bisect_lower, bisect_upper, maxiter=self.maxiter, xtol=self.xtol) # compute projection y = - mu * self.coeffs) return y.reshape(xshape)