from PEPit.function import Function
[docs]
class ConvexQGFunction(Function):
"""
The :class:`ConvexQGFunction` class overwrites the `add_class_constraints` method of :class:`Function`,
implementing the interpolation constraints of the class of quadratically upper bounded (:math:`\\text{QG}^+` [1]),
i.e. :math:`\\forall x, f(x) - f_\\star \\leqslant \\frac{L}{2} \\|x-x_\\star\\|^2`, and convex functions.
Attributes:
L (float): The quadratic upper bound parameter
General quadratically upper bounded (:math:`\\text{QG}^+`) convex functions are characterized
by the quadratic growth parameter `L`, hence can be instantiated as
Example:
>>> from PEPit import PEP
>>> from PEPit.functions import ConvexQGFunction
>>> problem = PEP()
>>> func = problem.declare_function(function_class=ConvexQGFunction, L=1)
References:
`[1] B. Goujaud, A. Taylor, A. Dieuleveut (2022).
Optimal first-order methods for convex functions with a quadratic upper bound.
<https://arxiv.org/pdf/2205.15033.pdf>`_
"""
def __init__(self,
L,
is_leaf=True,
decomposition_dict=None,
reuse_gradient=False,
name=None):
"""
Args:
L (float): The quadratic upper bound 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`.
"""
super().__init__(is_leaf=is_leaf,
decomposition_dict=decomposition_dict,
reuse_gradient=reuse_gradient,
name=name,
)
# Store L
self.L = L
[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 = (fi - fj >= gj * (xi - xj))
return constraint
[docs]
def set_qg_convexity_constraint_i_j(self,
xi, gi, fi,
xj, gj, fj,
):
"""
Formulates the list of interpolation constraints for self (qg convex function).
"""
# Interpolation conditions of QG convex functions class
constraint = (fi - fj >= gj * (xi - xj) + 1 / (2 * self.L) * gj ** 2)
return constraint
[docs]
def add_class_constraints(self):
"""
Formulates the list of interpolation constraints for self (quadratically maximally growing convex function);
see [1, Theorem 2.6].
"""
if self.list_of_stationary_points == list():
self.stationary_point()
self.add_constraints_from_two_lists_of_points(list_of_points_1=self.list_of_stationary_points,
list_of_points_2=self.list_of_points,
constraint_name="qg_convexity",
set_class_constraint_i_j=self.set_qg_convexity_constraint_i_j,
)
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,
)