  We propose an algorithm to calculate the exact solution for utility
optimization problems on finite state spaces under a class of
non-differentiable preferences. We prove that optimal strategies must lie on a
discrete grid in the plane, and this allows us to reduce the dimension of the
problem and define a very efficient method to obtain those strategies. We also
show how fast approximations for the value function can be obtained with an a
priori specified error bound and we use these to replicate results for
investment problems with a known closed-form solution. These results show the
efficiency of our approach, which can then be used to obtain numerical
solutions for problems for which no explicit formulas are known.
