from PEPit import PEP
from PEPit.operators import LipschitzOperator
[docs]
def wc_krasnoselskii_mann_increasing_step_sizes(n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the fixed point problem
.. math:: \\mathrm{Find}\\, x:\\, x = Ax,
where :math:`A` is a non-expansive operator, that is a :math:`L`-Lipschitz operator with :math:`L=1`.
This code computes a worst-case guarantee for the **Krasnolselskii-Mann** method. That is, it computes
the smallest possible :math:`\\tau(n)` such that the guarantee
.. math:: \\frac{1}{4}\\|x_n - Ax_n\\|^2 \\leqslant \\tau(n) \\|x_0 - x_\\star\\|^2
is valid, where :math:`x_n` is the output of the KM method, and :math:`x_\\star` is some fixed point of :math:`A`
(i.e., :math:`x_\\star=Ax_\\star`).
**Algorithm**: The KM method is described by
.. math:: x_{t+1} = \\frac{1}{t + 2} x_{t} + \\left(1 - \\frac{1}{t + 2}\\right) Ax_{t}.
**Reference**: This scheme was first studied using PEPs in [1].
`[1] F. Lieder (2018).
Projection Based Methods for Conic Linear Programming
Optimal First Order Complexities and Norm Constrained Quasi Newton Methods.
PhD thesis, HHU Düsseldorf.
<https://docserv.uni-duesseldorf.de/servlets/DerivateServlet/Derivate-49971/Dissertation.pdf>`_
Args:
n (int): number of iterations.
wrapper (str): the name of the wrapper to be used.
solver (str): the name of the solver the wrapper should use.
verbose (int): level of information details to print.
- -1: No verbose at all.
- 0: This example's output.
- 1: This example's output + PEPit information.
- 2: This example's output + PEPit information + solver details.
Returns:
pepit_tau (float): worst-case value
theoretical_tau (None): no theoretical value
Example:
>>> pepit_tau, theoretical_tau = wc_krasnoselskii_mann_increasing_step_sizes(n=3, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 6x6
(PEPit) Setting up the problem: performance measure is the minimum of 1 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (1 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
Function 1 : Adding 10 scalar constraint(s) ...
Function 1 : 10 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.11963370896691235
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 1.146590974387308e-09
All the primal scalar constraints are verified up to an error of 6.733909263534343e-10
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual scalar values associated with inequality constraints are nonnegative
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 1.650803029863701e-08
(PEPit) Final upper bound (dual): 0.11963371048428333 and lower bound (primal example): 0.11963370896691235
(PEPit) Duality gap: absolute: 1.5173709788651735e-09 and relative: 1.2683473512342912e-08
*** Example file: worst-case performance of Kranoselskii-Mann iterations ***
PEPit guarantee: 1/4 ||xN - AxN||^2 <= 0.119634 ||x0 - x_*||^2
"""
# Instantiate PEP
problem = PEP()
# Declare a non expansive operator
A = problem.declare_function(LipschitzOperator, L=1.)
# Start by defining its unique optimal point xs = x_*
xs, _, _ = A.fixed_point()
# Then define the starting point x0 of the algorithm
x0 = problem.set_initial_point()
# Set the initial constraint that is the difference between x0 and xs
problem.set_initial_condition((x0 - xs) ** 2 <= 1)
x = x0
for i in range(n):
x = 1 / (i + 2) * x + (1 - 1 / (i + 2)) * A.gradient(x)
# Set the performance metric to distance between xN and AxN
problem.set_performance_metric((1 / 2 * (x - A.gradient(x))) ** 2)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(wrapper=wrapper, solver=solver, verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
theoretical_tau = None
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of Kranoselskii-Mann iterations ***')
print('\tPEPit guarantee:\t 1/4 ||xN - AxN||^2 <= {:.6} ||x0 - x_*||^2'.format(pepit_tau))
# Return the worst-case guarantee of the evaluated method (and the reference theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
pepit_tau, theoretical_tau = wc_krasnoselskii_mann_increasing_step_sizes(n=3,
wrapper="cvxpy", solver=None,
verbose=1)