  The linear optimization algorithm ART3+O introduced by Chen et al. (2010) can
efficiently solve large scale inverse planning problems encountered in
radiation therapy by iterative projection. Its major weakness is that it cannot
guarantee ${\epsilon}$-optimality of the final solution due to an arbitrary
stopping criterion. We propose an improvement to ART3+O where the stopping
criterion is based on Farkas' lemma. The same theory can be used to detect
inconsistency in other projection methods as well. The proposed algorithm
guarantees to find an ${\epsilon}$-optimal solution in finite time. The
algorithm is demonstrated on numerical examples in radiation therapy.
