The Multi-path Utility Maximization and Multi-path TCP Design
  The network utility maximization problem (NUM) for multi-path is a problem which is non-strictly convex and non-separable. Using Jensen's inequality, we approximate the NUM to a strictly convex and separable problem which can be solved efficiently by the dual decomposition method. After a series of approximations, the result of the approximation problem converges to the globally optimal solution of the original problem.   Moreover, because of the separable and dual-based natures of the proposed algorithm, we utilize the reverse engineering frameworks of the current TCPs to develop a series of multi-path TCPs which are totally compatible with current TCPs. The multi-path users using our protocols can run simultaneously with the single-path users using the current TCPs. The simulations of our Multi-path Reno on ns-2 show the compatibility and the fairness among multi-path and single-path users. 
Phuong L. Vo, Anh T. Le, Choong S. Hong
math.OC