We evaluate and extend the solution methods for models with binary and multiple continuous choice variables in dynamic programming, particularly in cases where a discrete state space solution method is not viable. Therefore, we approximate the solution using taste shocks or black-box optimizers that applied mathematicians use to benchmark their algorithms. We apply these methods to a default framework in which agents have to solve a portfolio problem with long-term debt. We show that the choice of solution method matters, as taste shocks fail to attain convergence in multidimensional problems. We compare the relative advantages of using four optimization algorithms: the Nelder-Mead downhill simplex algorithm, Powell's direction-set algorithm with LINMIN, the conjugate gradient method BOBYQA, and the quasi-Newton Davidon-Fletcher-Powell (DFPMIN) algorithm. All of these methods, except for the last one, are preferred when derivatives cannot be easily computed. Ultimately, we find that Powell's routine evaluated with B-splines, while slow, is the most viable option. BOBYQA came in second place, while the other two methods performed poorly.