We study the problem of computing the capacity of a discrete memoryless channel under uncertainty affecting the channel law matrix, and possibly with a constraint on the average cost of the input distribution. The problem has been formulated in the literature as a max-min problem. We use the robust optimization methodology to convert the max-min problem to a standard convex optimization problem. For small-sized problems, and for many types of uncertainty, such a problem can be solved in principle using interior point methods (IPM). However, for large-scale problems, IPM are not practical. Here, we suggest an $\mathcal{O}(1/T)$ first-order algorithm based on Nemirovski (2004) which is applied directly to the max-min problem.