In this paper we develop an algorithm to find the maximum likelihood estimator of a convex hazard function. The maximization is done in two steps. First, we use the support reduction algorithm of [GJW1] to find the profile likelihood over a constrained space. We next show that (−1) times the profile likelihood is bathtub-shaped in the parameters, and use a bisection algorithm to find the overall maximizer. We use the same approach to find a least squares estimator of a convex hazard rate. Simulations and data examples are also given.