Approximate Douglas-Rachford algorithm for two-sets convex feasibility problems

TL;DR

This paper introduces the Approximate Douglas-Rachford algorithm, which combines DR and Frank-Wolfe methods to efficiently solve convex feasibility problems using inexact projections.

Contribution

The paper proposes a novel algorithm that integrates DR and Frank-Wolfe methods, enabling inexact projections and convergence analysis for convex feasibility problems.

Findings

Main sequence converges to a fixed point of DR operator

Shadow sequence converges to the feasible set

Numerical experiments illustrate algorithm behavior

Abstract

In this paper, we propose a new algorithm combining the Douglas-Rachford (DR) algorithm and the Frank-Wolfe algorithm, also known as the conditional gradient (CondG) method, for solving the classic convex feasibility problem. Within the algorithm, which will be named {\it Approximate Douglas-Rachford (ApDR) algorithm}, the CondG method is used as a subroutine to compute feasible inexact projections on the sets under consideration, and the ApDR iteration is defined based on the DR iteration. The ApDR algorithm generates two sequences, the main sequence, based on the DR iteration, and its corresponding shadow sequence. When the intersection of the feasible sets is nonempty, the main sequence converges to a fixed point of the usual DR operator, and the shadow sequence converges to the solution set. We provide some numerical experiments to illustrate the behaviour of the sequences produced by the proposed algorithm.