A Parallel Inertial Proximal Optimization Method

The Douglas-Rachford algorithm is a popular iterative method for finding a zero of a sum of two maximally monotone operators defined on a Hilbert space. In this paper, we propose an extension of this algorithm including inertia parameters and develop parallel versions to deal with the case of a sum of an arbitrary number of maximal operators. Based on this algorithm, parallel proximal algorithms are proposed to minimize over a linear subspace of a Hilbert space the sum of a finite number of proper, lower semicontinuous convex functions composed with linear operators. It is shown that particular cases of these methods are the simultaneous direction method of multipliers proposed by Stetzer et al., the parallel proximal algorithm developed by Combettes and Pesquet, and a parallelized version of an algorithm proposed by Attouch and Soueycatt.