import numpy as np
from PEPit.function import Function
[docs]
class LipschitzStronglyMonotoneOperator(Function):
"""
The :class:`LipschitzStronglyMonotoneOperator` class overwrites the `add_class_constraints` method
of :class:`Function`, implementing some constraints (which are not necessary and sufficient for interpolation)
for the class of Lipschitz continuous strongly monotone (and maximally monotone) operators.
Note:
Operator values can be requested through `gradient` and `function values` should not be used.
Warning:
Lipschitz strongly monotone operators do not enjoy known interpolation conditions. The conditions implemented
in this class are necessary but a priori not sufficient for interpolation. Hence, the numerical results
obtained when using this class might be non-tight upper bounds (see Discussions in [1, Section 2]).
Attributes:
mu (float): strong monotonicity parameter
L (float): Lipschitz parameter
Lipschitz continuous strongly monotone operators are characterized by parameters :math:`\\mu` and `L`,
hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.operators import LipschitzStronglyMonotoneOperator
>>> problem = PEP()
>>> h = problem.declare_function(function_class=LipschitzStronglyMonotoneOperator, mu=.1, L=1.)
References:
`[1] E. Ryu, A. Taylor, C. Bergeling, P. Giselsson (2020).
Operator splitting performance estimation: Tight contraction factors and optimal parameter selection.
SIAM Journal on Optimization, 30(3), 2251-2271.
<https://arxiv.org/pdf/1812.00146.pdf>`_
"""
def __init__(self,
mu,
L,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=True,
name=None):
"""
Args:
mu (float): The strong monotonicity parameter.
L (float): The Lipschitz continuity parameter.
is_leaf (bool): True if self is defined from scratch.
False if self is defined as linear combination of leaf .
decomposition_dict (dict): Decomposition of self as linear combination of leaf :class:`Function` objects.
Keys are :class:`Function` objects and values are their associated coefficients.
reuse_gradient (bool): If True, the same subgradient is returned
when one requires it several times on the same :class:`Point`.
If False, a new subgradient is computed each time one is required.
name (str): name of the object. None by default. Can be updated later through the method `set_name`.
Note:
Lipschitz continuous strongly monotone operators are necessarily continuous,
hence `reuse_gradient` is set to True.
"""
super().__init__(is_leaf=is_leaf,
decomposition_dict=decomposition_dict,
reuse_gradient=True,
name=name,
)
# Store L and mu
self.mu = mu
self.L = L
if self.L == np.inf:
print("\033[96m(PEPit) The class of Lipschitz strongly monotone operators is necessarily continuous.\n"
"To instantiate an operator, please avoid using the class LipschitzStronglyMonotoneOperator with\n"
" L == np.inf. Instead, please use the class StronglyMonotoneOperator (which accounts for the fact\n"
"that the image of the operator at certain points might not be a singleton).\033[0m")
[docs]
def set_strong_monotonicity_constraint_i_j(self,
xi, gi, fi,
xj, gj, fj,
):
"""
Set strong monotonicity constraint for operators.
"""
# Set constraint
constraint = ((gi - gj) * (xi - xj) - self.mu * (xi - xj)**2 >= 0)
return constraint
[docs]
def set_lipschitz_continuity_constraint_i_j(self,
xi, gi, fi,
xj, gj, fj,
):
"""
Set Lipschitz continuity constraint for operators.
"""
# Set constraint
constraint = ((gi - gj) ** 2 - self.L ** 2 * (xi - xj) ** 2 <= 0)
return constraint
[docs]
def add_class_constraints(self):
"""
Formulates the list of necessary conditions for interpolation of self (Lipschitz strongly monotone and
maximally monotone operator), see, e.g., discussions in [1, Section 2].
"""
self.add_constraints_from_two_lists_of_points(list_of_points_1=self.list_of_points,
list_of_points_2=self.list_of_points,
constraint_name="strong_monotonicity",
set_class_constraint_i_j=
self.set_strong_monotonicity_constraint_i_j,
symmetry=True,
)
self.add_constraints_from_two_lists_of_points(list_of_points_1=self.list_of_points,
list_of_points_2=self.list_of_points,
constraint_name="lipschitz_continuity",
set_class_constraint_i_j=
self.set_lipschitz_continuity_constraint_i_j,
symmetry=True,
)