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Traffic management is the adaptation of source rates and routing to efficiently utilize network resources. Recently, the complicated interactions between different Internet traffic management modules have been elegantly modeled by distributed primal-dual utility maximization, which sheds new light for developing effective management protocols. For single-path routing with given routes, the dual is a strictly concave network optimization problem. Unfortunately, the general form of multipath utility optimization is not strictly concave, making its solution quite unstable. Decomposition-based techniques like TRaffic-management Using Multipath Protocol (TRUMP) alleviates the instability, but their convergence is not guaranteed, nor is their optimality. They are also inflexible in differentiating the control at different links. In this paper, we address the above issues through a novel logarithm-barrier-based approach. Our approach jointly considers user utility and routing/congestion control. It translates the multipath utility maximization into a sequence of unconstrained optimization problems, with infinite logarithm barriers being deployed at the constraint boundary. We demonstrate that setting up barriers is much simpler than choosing traditional cost functions and, more importantly, it makes optimal solution achievable. We further demonstrate a distributed implementation, together with the design of a practical Logarithm Barrier-based-Multipath Protocol (LBMP). We evaluate the performance of LBMP through both numerical analysis and packet-level simulations. The results show that LBMP achieves high throughput and fast convergence over diverse representative network topologies. Such performance is comparable to TRUMP, and is often better. Moreover, LBMP is flexible in differentiating the control at different links, and its optimality and convergence are theoretically guaranteed.