Effectively solving many inverse problems in engineering requires to leverage all possible prior information about the structure of the signal to be estimated. This often leads to tackling constrained optimization problems with mixtures of regularizers. Providing a general purpose optimization algorithm for these cases, with both guaranteed convergence rate as well as fast implementation remains an important challenge. In this paper, we describe how a recent primaldual algorithm for non-smooth constrained optimization can be successfully used to tackle these problems. Its simple iterations can be easily parallelized, allowing very efficient computations. Furthermore, the algorithm is guaranteed to achieve an optimal convergence rate for this class of problems. We illustrate its performance on two problems, a compressive magnetic resonance imaging application and an approach for improving the quality of analog-to-digital conversion of amplitude-modulated signals.