Bierlaire, M., Thémans, M., and Zufferey, N. (2007)

A heuristic for nonlinear global optimization relevant to discrete choice models estimation

of the Sixth Triennial Symposium on Transportation Analysis (TRISTAN), Phuket Island, Thailand

In most applications related to transportation, it is of major importance to be able to identify the global optimum of the associated optimization problem. The work we present in this paper is motivated by the optimization problems arising in the maximum likelihood estimation of discrete choice models. Estimating those models becomes more and more problematic as several issues may occur in the estimation. We focus our interest on the non-concavity of the log-likelihood function which can present several (and often many) local optima in the case of advanced models. In this context, we propose a new heuristic for nonlinear global optimization combining a variable neighborhood search framework with a modified trust-region algorithm as local search. The proposed method presents the capability to prematurely interrupt the local search if the iterates are converging to a local minimum which has already been visited or if they are reaching an area where no significant improvement can be expected. The neighborhoods as well as the neighbors selection procedure are exploiting the curvature of the objective function. Numerical tests are performed on a set of unconstrained nonlinear problems from the literature. Results illustrate that the new method significantly outperforms existing heuristics from the literature in terms of success rate, CPU time, and number of function evaluations.