import numpy as np
from PEPit.function import Function
[docs]class SmoothFunction(Function):
"""
The :class:`SmoothFunction` class overwrites the `add_class_constraints` method of :class:`Function`,
implementing the interpolation constraints of the class of smooth (not necessarily convex) functions.
Attributes:
L (float): smoothness parameter
Smooth functions are characterized by the smoothness parameter `L`, hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.functions import SmoothFunction
>>> problem = PEP()
>>> func = problem.declare_function(function_class=SmoothFunction, L=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,
L=1.,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=True):
"""
Args:
L (float): The smoothness 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.
Note:
Smooth functions are necessarily differentiable, hence `reuse_gradient` is set to True.
"""
super().__init__(is_leaf=is_leaf,
decomposition_dict=decomposition_dict,
reuse_gradient=True)
# Store L
self.L = L
if self.L == np.inf:
print("\033[96m(PEPit) The class of L-smooth functions with L == np.inf implies no constraint: \n"
"it contains all differentiable functions. This might imply issues in your code.\033[0m")
[docs] def add_class_constraints(self):
"""
Formulates the list of interpolation constraints for self (smooth (not necessarily convex) function),
see [1, Theorem 3.10].
"""
for point_i in self.list_of_points:
xi, gi, fi = point_i
for point_j in self.list_of_points:
xj, gj, fj = point_j
if point_i != point_j:
# Interpolation conditions of smooth functions class
self.list_of_class_constraints.append(fi - fj
- self.L/4 * (xi - xj)**2
- 1/2 * (gi + gj) * (xi - xj)
+ 1/(4 * self.L) * (gi - gj)**2
<= 0)