Development of innovative techniques in nonlinear and stochastic programming and applications in discrete choice theory

本研究的目的是设计非凸随机规划的方法，应用于工业和工程中的实际问题，例如交通运输的离散选择模型。第一个轴涉及非凸随机规划的自适应抽样方法的研究，基于传统的蒙特卡罗技术，但也更先进的战略，随机准蒙特卡罗方法。我们的目标是加快优化过程，同时确保开发的算法的一致性和收敛性。这些技术将应用于工程领域，特别是离散选择建模。第二个轴集中在优化过程中的结构开发。有趣的结构，事实上，往往存在于特定的情况下，随机程序，如问题来自混合logit估计和多级非凸随机规划。在第一种情况下，可以考虑特定的Hessian近似，以加速优化方法的收敛，就像最小二乘程序一样。在第二种情况下，它是可能的，表现出并行性，内点方法的上下文中。最后一个轴处理非参数估计，特别是混合logit模型，而这个想法可以扩展到各种随机程序，其中随机分布参数进行估计。然后，我们的目标是开发方法，避免任何先验假设的基础分布，而是允许任意分布的建设沿着的估计过程。

The proposed research aims to design methods for nonconvex stochastic programming, applied to practical problems arising in industry and engineering, for instance discrete choice modeling for transportation.This research follows three main axes. The first axis concerns the study of adaptive sampling methods for nonconvex stochastic programming, based on traditional Monte Carlo techniques, but also more advanced strategies as randomized quasi-Monte Carlo methods. The objective is to speed-up the optimization process, while ensuring consistency properties and convergence of the developed algorithms. These techniques will be applied in engineering field, in particular for discrete choice modeling. The second axis is focused on structure exploitation during the optimization procedure. Interesting structures, indeed, are often present in particular instances of stochastic programs, such as problems coming from mixed logit estimation and multistage nonconvex stochastic programming. In the first case, it is possible to consider specific Hessian approximations in order to speed-up the convergence of the optimization methods, as with least-squares programs. In the second case, it is possible to exhibit parallelism properties, in the context of interior point methods. The last axis treats nonparametric estimation, in particular for mixed logit models, while the idea can be extended to various stochastic programs where random distribution parameters are to be estimated. We then aim to develop methods that avoid any a priori assumptions on the underlying distributions, but instead allow the construction of arbitrary distributions along with the estimation process.