The \(\lambda \) -exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric point of view, but the development of computational methodologies is still in an early stage. In this paper, we propose a fixed point iteration for maximum likelihood estimation under i.i.d. sampling, and prove using the duality that the likelihood is monotone along the iterations. We illustrate the algorithm with the q-Gaussian distribution and the Dirichlet perturbation.