Using simulation to approximate subgradients of convex performance measures in service systems

We study the problem of approximating a subgradient of a convex (or concave) discrete function that is evaluated via simulation. This problem arises, for instance, in optimization problems such as finding the minimal cost staff schedule in a call center subject to a service level constraint. There, subgradient information can be used to significantly reduce the search space. The problem of approximating subgradients is closely related to the one of approximating gradients and we suggest and compare how three existing methods for computing gradients via simulation, i.e., finite differences, the likelihood ratio method and infinitesimal perturbation analysis, can be applied to approximate subgradients when the variables are discrete. We provide a computational study to highlight the properties of each approach.