Convergence theory for nonconvex stochastic programming with an application to mixed logit

Monte Carlo methods have been used extensively in the area of stochastic programming. As with other methods that involve a level of uncertainty, theoretical properties are required in order to give an indication of their performance. Traditional convergence properties of Monte Carlo methods in stochastic programming consider global minimisers or first-order critical points of the sample average approximation (SAA) problems as well as of the true problem. In this paper we review the first-order critical convergence and extend these results to the case where computed solutions are second-order critical, in turn allowing for problems whose objective function is nonconvex. As an application, we use the proposed framework in the estimation of mixed logit models for discrete choice, guaranteeing almost sure convergence of the solutions of the successive SAA problems. The result is observed to hold both for constrained and unconstrained problems. Finally, we produce estimates of the simulatio...

• Type: Journal
• Year: 2006
• Where: MP
• Authors: Fabian Bastin, Cinzia Cirillo, Philippe L. Toint
• Keywords: Monte Carlo, Monte Carlo Methods, MP 2006, Stochastic Programming