We show that the Bregman Douglas–Rachford splitting method (BDRS) can fail for matrix scaling and unregularized optimal transport. For matrix scaling, we construct a strictly positive \(36\times6\) integer matrix, positive rational marginals of equal mass, and a positive auxiliary initialization for which the matrix iterates enter a nonconstant six-cycle after one iteration. The construction prescribes rotating scaling directions and uses cyclic symmetry to reduce the required transitions to a single interpolation condition, which is linear in the block row masses. Exact rational arithmetic certifies the example, and a direct induction proves the cycle for all iterations. A block lifting produces a \(72\times12\) optimal transport counterexample with cost entries \(2\) and \(3\). For every fixed positive algorithm parameter, its iterates have six distinct subsequential limits and remain uniformly infeasible, although the transport problem has feasible solutions. These examples disprove the unconditional convergence of BDRS.
BDRS Can Fail for Matrix Scaling and Optimal Transport
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