In this paper, we study the computation of the rate-distortion-perception function (RDPF) for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint that belongs to the family of f-divergences. For that, we leverage the fact that RDPF, assuming mild regularity conditions on the perception constraint, forms a convex programming problem. We first develop parametric characterizations of the optimal solution and utilize them in an alternating minimization approach for which we prove convergence guarantees. The resulting structure of the iterations of the alternating minimization approach renders the implementation of a generalized Blahut-Arimoto (BA) type of algorithm infeasible. To overcome this difficulty, we propose a relaxed formulation of the structure of the iterations in the alternating minimization approach, which allows for the implementation of an approximate iterative scheme. This approximation is shown, via the derivation of necessary and sufficient conditions, to guarantee convergence to a globally optimal solution. We also provide sufficient conditions on the distortion and the perception constraints which guarantee that our algorithm converges exponentially fast. We corroborate our theoretical results with numerical simulations, and we draw connections with existing results.