We provide an algorithm for solving multidimensional screening problems which are intractable analytically. The algorithm is a primal-dual algorithm which al- ternates between optimising the primal problem of the surplus extracted by the principal and the dual problem of the optimal assignment to deliver to the agents for a given surplus. We illustrate the algorithm by solving (i) the generic monopolist price discrimination problem and (ii) an optimal tax problem covering income and savings taxes when citizens differ in multiple dimensions.