SUMMARYOptimization theory and nonlinear programming method have successfully been applied into wire-lined networks (e.g., the Internet) in developing efficient resource allocation and congestion control schemes. The resource (e.g., bandwidth) allocation in a communication network has been modeled into an optimization problem: the objective is to maximize the source aggregate utility subject to the network resource constraint. However, for wireless networks, how to allocate the resource among the soft quality of service (QoS) traffic remains an important design challenge. Mathematically, the most difficult comes from the non-concave utility function of soft QoS traffic in the network utility maximization (NUM) problem. Previous result on this problem has only been able to find its sub-optimal solution. Facing this challenge, this paper establishes some key theorems to find the optimal solution and then present a complete algorithm called utility-based allocation for soft QoS to obtain the desired optimal solution. The proposed theorems and algorithm act as designing guidelines for resource allocation of soft QoS traffic in a wireless network, which take into account the total available resource of network, the users' traffic characteristics, and the users' channel qualities. By numerical examples, we illustrate the explicit solution procedures.