Coupled Adaptable Backward-Forward-Backward Resolvent Splitting Algorithm (CABRA): A Matrix-Parametrized Resolvent Splitting Method for the Sum of Maximal Monotone and Cocoercive Operators Composed with Linear Coupling Operators

TL;DR

This paper introduces a matrix-parametrized splitting algorithm for solving operator equations involving monotone and cocoercive operators, with applications in stochastic programming and decentralized optimization.

Contribution

It proposes a novel resolvent splitting method with matrix parameters, along with a semidefinite programming framework for optimal parameter selection.

Findings

Accelerates convergence through diagonal scaling and parallelization.

Demonstrates effectiveness in multi-stage stochastic programming.

Provides a decentralized approach with low data transfer requirements.

Abstract

We present a novel matrix-parametrized frugal splitting algorithm which finds the zero of a sum of maximal monotone and cocoercive operators composed with linear selection operators. We also develop a semidefinite programming framework for selecting matrix parameters and demonstrate its use for designing matrix parameters which provide beneficial diagonal scaling, allow parallelization, and adhere to a given communication structure. We show that taking advantage of the linear selection operators in this way accelerates convergence in numerical experiments, and show that even when the selection operators are the identity, we can accelerate convergence by using the matrix parameters to provide appropriately chosen diagonal scaling. We conclude by demonstrating the applicability of this algorithm to multi-stage stochastic programming, outlining a decentralized approach to the relaxed stochastic weapon target assignment problem which splits over the source nodes and has low data transfer and memory requirements.