  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.
