In spite of the growing consideration for optimal execution in the financial
mathematics literature, numerical approximations of optimal trading curves are
almost never discussed. In this article, we present a numerical method to
approximate the optimal strategy of a trader willing to unwind a large
portfolio. The method we propose is very general as it can be applied to
multi-asset portfolios with any form of execution costs, including a bid-ask
spread component, even when participation constraints are imposed. Our method,
based on convex duality, only requires Hamiltonian functions to have C1,1
regularity while classical methods require additional regularity and cannot be
applied to all cases found in practice