from PEPit.function import Function
[docs]class ConvexLipschitzFunction(Function):
"""
The :class:`ConvexLipschitzFunction` class overwrites the `add_class_constraints` method of :class:`Function`,
implementing the interpolation constraints of the class of convex closed proper (CCP) Lipschitz continuous functions.
Attributes:
M (float): Lipschitz parameter
CCP Lipschitz continuous functions are characterized by a parameter `M`, hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.functions import ConvexLipschitzFunction
>>> problem = PEP()
>>> func = problem.declare_function(function_class=ConvexLipschitzFunction, 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=1.,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=False):
"""
Args:
M (float): The Lipschitz continuity parameter of self.
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.
"""
# Inherit directly from Function.
super().__init__(is_leaf=is_leaf,
decomposition_dict=decomposition_dict,
reuse_gradient=reuse_gradient)
# param M
self.M = M
[docs] def add_class_constraints(self):
"""
Formulates the list of interpolation constraints for self (CCP Lipschitz continuous function),
see [1, Theorem 3.5].
"""
for point_i in self.list_of_points:
xi, gi, fi = point_i
# Lipschitz condition on the function (bounded gradient)
self.list_of_class_constraints.append(gi**2 <= self.M**2)
for point_j in self.list_of_points:
xj, gj, fj = point_j
if point_i != point_j:
# Interpolation conditions of convex functions class
self.list_of_class_constraints.append(fi - fj >= gj * (xi - xj))