The alternating direction method of multipliers (ADMM) has been widely used for a very wide variety of imaging inverse problems. One of the disadvantages of this method, however, is the need to select an algorithm parameter, the penalty parameter, that has a significant effect on the rate of convergence of the algorithm. Although a number of heuristic methods have been proposed, as yet there is no general theory providing a good choice of this parameter for all problems. One obvious approach would be to try a number of different parameters at each iteration, proceeding further with the one that delivers the best reduction in functional value, but this would involve a substantial increase in computational cost. We show that, when the forward operator of the inverse problem is represented via a transform operator, requiring iterative methods to solve the main linear system arising in the ADMM solution, it is possible to explore a large range of parameters at marginal additional cost, thus greatly improving the robustness of the method to the choice of penalty parameter.