A two-party quantum communication process with classical inputs and outcomes can be simulated by replacing the quantum channel with a classical one. The minimal amount of classical communication required to reproduce the statistics of the quantum process is called its communication complexity. In the case of many instances simulated in parallel, the minimal communication cost per instance is called the asymptotic communication complexity. Previously, we reduced the computation of the asymptotic communication complexity to a convex minimization problem. In most cases, the objective function does not have an explicit analytic form, as the function is defined as the maximum over an infinite set of convex functions. Therefore, the overall problem takes the form of a minimax problem and cannot directly be solved by standard optimization methods. In this paper, we introduce a simple algorithm to compute the asymptotic communication complexity. For some special cases with an analytic objective function one can employ available convex-optimization libraries. In the tested cases our method turned out to be notably faster. Finally, using our method we obtain 1.238 bits as a lower bound on the asymptotic communication complexity of a noiseless quantum channel with the capacity of 1 qubit. This improves the previous bound of 1.208 bits.