We consider the problem of demand estimation for public transport networks. Given an origin-destination matrix representing the public transport demand, the distribution of ow among different lines can be obtained assuming that it corresponds to a certain equilibrium characterized by an optimization problem. The knowledge of the origin-destination matrix is expensive and sometimes unaffordable in practice. Traditionally, it is estimated using statistical or econometrical considerations. In this work, we explore the estimation through the numerical solution of a bilevel optimization problem. One disadvantage of this formulation is the difficulty of obtaining descent directions, therefore we proposed a derivative-free method for the resolution of the optimization problem. The method is firstly tested on small networks using a derivative-free optimization method and then, using an approach based on simulation. This simulation-optimization methodology showed as good results as analytical modeling, opening the door to handle bigger networks where analytical computation is hard to accomplish.