Computing the resolvent of the sum of maximally monotone operators with the averaged alternating modified reflections algorithm

TL;DR

This work generalizes the averaged alternating modified reflections algorithm so that it can be used to compute the resolvent of the sum of two maximally monotone operators, which gives rise to a new splitting method, which is proved to be strongly convergent.

Abstract

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.

Figures (2)

• Figure 1: Illustration of the computation of the iterations of the splitting algorithms proposed, when they are applied to the normal cones of three balls $B_1,B_2,B_3\subset\mathbb{R}^2$, with $q=0$, $\lambda_n=0.4$ and $\beta=0.7$
• Figure 2: Results of the numerical experiment comparing the algorithms in \ref{['th:par_AAMR', 'th:par_AAMR2']}. In the top figure, we show the average number of iterations required by each algorithm with respect to the value of $\beta$. In the bottom figure, we show the ratio between the average number of iterations required by the original algorithm and the alterative one, for each number of constraints with respect to the value of $\beta$.

Theorems & Definitions (31)

• Definition 2.1
• Example 2.1: The subdifferential and the normal cone operators
• Lemma 2.1
• proof
• Definition 2.2
• proof
• Example 2.2: The proximity and the projector operators
• Definition 2.3
• Lemma 2.2
• proof