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# -*- coding: utf-8 -*-
# Copyright 2019 Science Technology Facilities Council
# Copyright 2019 University of Manchester
#
# This work is part of the Core Imaging Library developed by Science Technology
# Facilities Council and University of Manchester
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0.txt
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from ccpi.optimisation.algorithms import Algorithm
class PDHG(Algorithm):
r'''Primal Dual Hybrid Gradient
Problem:
.. math::
\min_{x} f(Kx) + g(x)
:param operator: Linear Operator = K
:param f: Convex function with "simple" proximal of its conjugate.
:param g: Convex function with "simple" proximal
:param sigma: Step size parameter for Primal problem
:param tau: Step size parameter for Dual problem
Remark: Convergence is guaranted provided that
.. math::
\tau \sigma \|K\|^{2} <1
Reference:
(a) A. Chambolle and T. Pock (2011), "A first-order primal–dual algorithm for convex
problems with applications to imaging", J. Math. Imaging Vision 40, 120–145.
(b) E. Esser, X. Zhang and T. F. Chan (2010), "A general framework for a class of first
order primal–dual algorithms for convex optimization in imaging science",
SIAM J. Imaging Sci. 3, 1015–1046.
'''
def __init__(self, f=None, g=None, operator=None, tau=None, sigma=1.,**kwargs):
'''PDHG algorithm creator
Optional parameters
:param operator: a Linear Operator
:param f: Convex function with "simple" proximal of its conjugate.
:param g: Convex function with "simple" proximal
:param sigma: Step size parameter for Primal problem
:param tau: Step size parameter for Dual problem
'''
super(PDHG, self).__init__(**kwargs)
if f is not None and operator is not None and g is not None:
self.set_up(f=f, g=g, operator=operator, tau=tau, sigma=sigma)
def set_up(self, f, g, operator, tau=None, sigma=1.):
'''initialisation of the algorithm
:param operator: a Linear Operator
:param f: Convex function with "simple" proximal of its conjugate.
:param g: Convex function with "simple" proximal
:param sigma: Step size parameter for Primal problem
:param tau: Step size parameter for Dual problem'''
print("{} setting up".format(self.__class__.__name__, ))
# can't happen with default sigma
if sigma is None and tau is None:
raise ValueError('Need sigma*tau||K||^2<1')
# algorithmic parameters
self.f = f
self.g = g
self.operator = operator
self.tau = tau
self.sigma = sigma
if self.tau is None:
# Compute operator Norm
normK = self.operator.norm()
# Primal & dual stepsizes
self.tau = 1 / (self.sigma * normK ** 2)
self.x_old = self.operator.domain_geometry().allocate()
self.x_tmp = self.x_old.copy()
self.x = self.x_old.copy()
self.y_old = self.operator.range_geometry().allocate()
self.y_tmp = self.y_old.copy()
self.y = self.y_old.copy()
self.xbar = self.x_old.copy()
# relaxation parameter
self.theta = 1
self.update_objective()
self.configured = True
print("{} configured".format(self.__class__.__name__, ))
def update(self):
# save previous iteration
self.x_old.fill(self.x)
self.y_old.fill(self.y)
# Gradient ascent for the dual variable
self.operator.direct(self.xbar, out=self.y_tmp)
# self.y_tmp *= self.sigma
# self.y_tmp += self.y_old
self.y_tmp.axpby(self.sigma, 1 , self.y_old, self.y_tmp)
# self.y = self.f.proximal_conjugate(self.y_old, self.sigma)
self.f.proximal_conjugate(self.y_tmp, self.sigma, out=self.y)
# Gradient descent for the primal variable
self.operator.adjoint(self.y, out=self.x_tmp)
# self.x_tmp *= -1*self.tau
# self.x_tmp += self.x_old
self.x_tmp.axpby(-self.tau, 1. , self.x_old, self.x_tmp)
self.g.proximal(self.x_tmp, self.tau, out=self.x)
# Update
self.x.subtract(self.x_old, out=self.xbar)
# self.xbar *= self.theta
# self.xbar += self.x
self.xbar.axpby(self.theta, 1 , self.x, self.xbar)
def update_objective(self):
p1 = self.f(self.operator.direct(self.x)) + self.g(self.x)
d1 = -(self.f.convex_conjugate(self.y) + self.g.convex_conjugate(-1*self.operator.adjoint(self.y)))
self.loss.append([p1, d1, p1-d1])
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