import numpy as np
from PEPit.function import Function
[docs]
class ConvexSupportFunction(Function):
"""
The :class:`ConvexSupportFunction` class overwrites the `add_class_constraints` method of :class:`Function`,
implementing interpolation constraints for the class of closed convex support functions.
Attributes:
M (float): upper bound on the Lipschitz constant
Convex support functions are characterized by a parameter `M`, hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.functions import ConvexSupportFunction
>>> problem = PEP()
>>> func = problem.declare_function(function_class=ConvexSupportFunction, M=1)
References:
`[1] A. Taylor, J. Hendrickx, F. Glineur (2017).
Exact worst-case performance of first-order methods for composite convex optimization.
SIAM Journal on Optimization, 27(3):1283–1313.
<https://arxiv.org/pdf/1512.07516.pdf>`_
"""
def __init__(self,
M=np.inf,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=False,
name=None):
"""
Args:
M (float): Lipschitz constant of self. Default value set to infinity.
is_leaf (bool): True if self is defined from scratch.
False is 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`.
"""
super().__init__(is_leaf=is_leaf,
decomposition_dict=decomposition_dict,
reuse_gradient=reuse_gradient,
name=name,
)
# Store the Lipschitz constant in an attribute
self.M = M
[docs]
@staticmethod
def set_fenchel_value_constraint_i(xi, gi, fi):
"""
Set the value of the Fenchel transform to 0.
"""
# Set constraint
constraint = (gi * xi - fi == 0)
return constraint
[docs]
def set_lipschitz_continuity_constraint_i(self, xi, gi, fi):
"""
Set Lipschitz continuity constraint so that its Fenchel transform has a bounded support.
"""
# Set constraint
constraint = (gi ** 2 <= self.M ** 2)
return constraint
[docs]
@staticmethod
def set_convexity_constraint_i_j(xi, gi, fi,
xj, gj, fj,
):
"""
Formulates the list of interpolation constraints for self (CCP function).
"""
# Interpolation conditions of convex functions class
constraint = (xj * (gi - gj) <= 0)
return constraint
[docs]
def add_class_constraints(self):
"""
Formulates the list of interpolation constraints for self (closed convex support function),
see [1, Corollary 3.7].
"""
self.add_constraints_from_one_list_of_points(list_of_points=self.list_of_points,
constraint_name="fenchel_value",
set_class_constraint_i=self.set_fenchel_value_constraint_i,
)
if self.M != np.inf:
self.add_constraints_from_one_list_of_points(list_of_points=self.list_of_points,
constraint_name="lipschitz_continuity",
set_class_constraint_i=
self.set_lipschitz_continuity_constraint_i,
)
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="convexity",
set_class_constraint_i_j=self.set_convexity_constraint_i_j,
)