  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.
