import numpy as np
from PEPit.function import Function
[docs]
class LipschitzOperator(Function):
"""
The :class:`LipschitzOperator` class overwrites the `add_class_constraints` method of :class:`Function`,
implementing the interpolation constraints of the class of Lipschitz continuous operators.
Note:
Operator values can be requested through `gradient` and `function values` should not be used.
Attributes:
L (float): Lipschitz parameter
Cocoercive operators are characterized by the parameter :math:`L`, hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.operators import LipschitzOperator
>>> problem = PEP()
>>> func = problem.declare_function(function_class=LipschitzOperator, L=1.)
Notes:
By setting L=1, we define a non-expansive operator.
By setting L<1, we define a contracting operator.
References:
[1] M. Kirszbraun (1934).
Uber die zusammenziehende und Lipschitzsche transformationen.
Fundamenta Mathematicae, 22 (1934).
[2] F.A. Valentine (1943).
On the extension of a vector function so as to preserve a Lipschitz condition.
Bulletin of the American Mathematical Society, 49 (2).
[3] F.A. Valentine (1945).
A Lipschitz condition preserving extension for a vector function.
American Journal of Mathematics, 67(1).
Discussions and appropriate pointers for the interpolation problem can be found in:
`[4] 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,
L,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=True,
name=None):
"""
Args:
L (float): 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 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
self.L = L
if self.L == np.inf:
print("\033[96m(PEPit) The class of L-Lipschitz operators with L == np.inf implies no constraint: \n"
"it contains all multi-valued mappings. This might imply issues in your code.\033[0m")
[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 interpolation constraints for self (Lipschitz operator),
see [1, 2, 3] or e.g., [4, Fact 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="lipschitz_continuity",
set_class_constraint_i_j=
self.set_lipschitz_continuity_constraint_i_j,
symmetry=True,
)