# On the behaviour of the Douglas-Rachford algorithm for minimizing a convex function subject to a linear constraint

## TL;DR

This paper investigates the Douglas-Rachford algorithm's behavior when minimizing a convex function under a linear constraint, including cases with no common feasible points, and introduces new convergence results and parallel splitting methods.

## Contribution

It provides new convergence results for the DRA in the absence of feasible points and introduces a novel parallel splitting approach for constrained convex minimization.

## Key findings

- DRA converges to a best approximation solution even without feasible points.
- New parallel splitting method for convex optimization with linear constraints.
- Illustrative examples demonstrating theoretical results.

## Abstract

The Douglas-Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples.