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.
