In this paper, we are interested in the recovery of an unknown signal corrupted by a linear operator, a nonlinear function, and an additive Gaussian noise. In addition, some of the observations contain outliers. Many robust data fit functions which alleviate sensitivity to outliers can be expressed as piecewise rational functions. Based on this fact, we reformulate the robust inverse problem as a rational optimization problem. The considered framework allows us to incorporate nonconvex constraints such as unions of subsets. The rational problem is then solved using recent optimization techniques which offer guarantees for global optimality. Finally, experimental results illustrate the validity of the recovered global solutions and the good quality of the reconstructed signals despite the presence of outliers.