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

本文的研究目标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建模。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方法，但也有更先进的策略，如随机准蒙特卡罗方法。目标是加速优化过程，同时确保所开发算法的一致性和收敛。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。实际上，有趣的结构经常出现在随机程序的特定实例中，例如来自混合Logit估计和多阶段非凸随机规划的问题。在第一种情况下，可以考虑特定的海森近似，以加快优化方法的收敛速度，就像最小二乘程序一样。在第二种情况下，可以在内点方法的上下文中展示并行性。最后一个轴处理非参数估计，特别是对于混合Logit模型，而这个想法可以扩展到各种随机程序，其中随机分布参数被估计。然后我们的目标是开发方法，避免对潜在分布的任何先验假设，而是允许在估计过程中构建任意分布。

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.