We consider a multimedia transmission scheme that is based on the combination of layered source coding with successive refinement and on using a broadcast strategy to send the layers over Rayleigh fading channels. We optimize the power allocated to each layer in order to maximize the average user satisfaction defined by a utility function of the total decoded rate. This optimization problem is non-convex, and hence difficult to solve. However, in this paper, we build up on our recently proposed mathematical framework, and we describe, as the main contribution, an efficient algorithm to solve this problem. Using numerical examples, we show that significant gains in the outage probability can be achieved by applying our proposed algorithm.