In the last ?fteen years a signi?cant progress was achieved by considering an entropic relaxation of the classical multi-partite optimal transport prob-lem (MPOTP). The entropic relaxation gives rise to the rescaling problem of a given tensor. This rescaling can be achieved fast with the Sinkhorn type algorithms. Recently, it was shown that a similar approach works for the quan-tum MPOTP. However, the analog of the rescaling Sinkhorn algorithm is much more complicated than in the classical MTOTP. In this talk we show that the interior point method (IPM) for the primary and dual problems of classical and quantum MPOTP problems can be considered as barrier relaxations of the optimal transport problems (OTP). It is well known that the dual of the OTP are advantageous as it has much less variables than the primary problem. The IPM for the dual problem of the classical MPOTP are not as fast as the Sinkhorn type algorithm. However, IPM method for the dual of the quantum MPOTP seems to work quite e?ciently.
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