The Douglas–Rachford method: Maximally monotone + strongly monotone/cocoercive

Problem setup

Consider the monotone inclusion

\[\text{find } x \in \calH \text{ such that } 0 \in G_1(x) + G_2(x),\]

where:

• \(G_1 : \calH \rightrightarrows \calH\) is maximally monotone,

• \(G_2 : \calH \to \calH\) is both \(\mu\)-strongly monotone and \(\beta\)-cocoercive, with \(\mu>0\) and \(\beta>0\).

For an initial point \(x^0 \in \calH\), step size \(\gamma \in \reals_{++}\), and relaxation \(\lambda \in \reals\), the Douglas–Rachford update is

\[\begin{split}(\forall k \in \naturals)\quad \left[ \begin{aligned} v^k &= J_{\gamma G_1}(x^k), \\ w^k &= J_{\gamma G_2}(2v^k - x^k), \\ x^{k+1} &= x^k + \lambda (w^k - v^k). \end{aligned} \right.\end{split}\]

In this example, we search for the smallest contraction factor \(\rho\in[0,1)\) provable using AutoLyap such that

\[\|x^k - x^\star\|^2 = O(\rho^k) \quad \textup{ as } \quad k\to\infty,\]

where

\[x^\star \in \zer(G_1 + G_2).\]

Model the problem in AutoLyap and search for the smallest rho

• \(G_1\) is modeled by MaximallyMonotone.

• \(G_2\) is modeled as an intersection of StronglyMonotone and Cocoercive.

• The full inclusion is built with InclusionProblem.

• The update rule is represented by DouglasRachford.

• Distance-to-solution parameters are obtained with IterationIndependent.LinearConvergence.get_parameters_distance_to_solution.

• The contraction factor is searched with IterationIndependent.LinearConvergence.bisection_search_rho.

Run the iteration-independent analysis

This example uses the MOSEK Fusion backend (backend="mosek_fusion"). Install the optional MOSEK dependency first:

pip install "autolyap[mosek]"

import numpy as np

from autolyap import SolverOptions
from autolyap.algorithms import DouglasRachford
from autolyap.problemclass import (
    Cocoercive,
    InclusionProblem,
    MaximallyMonotone,
    StronglyMonotone,
)
from autolyap.iteration_independent import IterationIndependent

mu = 1.0
beta = 0.5
gamma = 1.0
lambda_value = 2.0

problem = InclusionProblem(
    [
        MaximallyMonotone(),  # G1
        [StronglyMonotone(mu=mu), Cocoercive(beta=beta)],  # G2 in intersection
    ]
)
algorithm = DouglasRachford(gamma=gamma, lambda_value=lambda_value, type="operator")
solver_options = SolverOptions(backend="mosek_fusion")
# License-free option:
# solver_options = SolverOptions(backend="cvxpy", cvxpy_solver="CLARABEL")

# Build V(P, p, k) and R(T, t, k) for distance-to-solution analysis.
P, T = IterationIndependent.LinearConvergence.get_parameters_distance_to_solution(
    algorithm
)

result = IterationIndependent.LinearConvergence.bisection_search_rho(
    problem,
    algorithm,
    P,
    T,
    S_equals_T=True,
    s_equals_t=True,
    remove_C3=True,
    solver_options=solver_options,
)

if result["status"] != "feasible":
    raise RuntimeError("No feasible Lyapunov certificate in the requested rho interval.")

rho_autolyap = result["rho"]

alpha = lambda_value / 2.0
delta = np.sqrt(
    1.0 - (4.0 * gamma * mu) / (1.0 + 2.0 * gamma * mu + (gamma ** 2) * mu / beta)
)
rho_theory = (abs(1.0 - alpha) + alpha * delta) ** 2

print(f"rho (AutoLyap): {rho_autolyap:.8f}")
print(f"rho (theory):   {rho_theory:.8f}")

What to inspect in result:

• result["status"]: feasible, infeasible, or not_solved.

• result["solve_status"]: raw backend status.

• result["rho"]: certified contraction factor when feasible.

• result["certificate"]: Lyapunov certificate matrices/scalars.

The computed value rho (AutoLyap) matches (up to solver numerical tolerances) the theoretical rate expression in [Gis17, Theorem 7.4], i.e.,

\[\|x^k - x^\star\|^2 = O(\rho^k) \quad \textup{ as } \quad k\to\infty,\]

where

\[\rho = \left(|1-\alpha| + \alpha \delta\right)^2, \quad \alpha = \frac{\lambda}{2}, \quad \delta = \sqrt{1 - \frac{4\gamma\mu}{1 + 2\gamma\mu + \gamma^2 \mu / \beta}},\]

and

\[x^\star \in \zer(G_1 + G_2).\]

Sweeping over 100 values of \(\gamma\) on \(0 < \gamma \le 5\) (with \(\mu=1\), \(\beta=0.5\), \(\lambda=2\)) gives the plot below, with the theoretical rate in black and AutoLyap certificates as blue dots.

References

[Gis17]

Pontus Giselsson. Tight global linear convergence rate bounds for Douglas–Rachford splitting. Journal of Fixed Point Theory and Applications, 19(4):2241–2270, 2017. doi:10.1007/s11784-017-0417-1.