Towards new solution techniques in mathematical programming with scenarios

运筹学目前在科研和工业领域占有重要的地位。随着计算能力的发展，研究人员和实践者越来越关注操作过程的优化，即预测模型的估计。然而，这样的模型存在许多潜在的简化，其中一个主要的简化是假设所有参数都是完全已知的。不幸的是，这很少是真的，在数学问题中融入不确定性的兴趣越来越大。这允许产生更灵活和更强大的解决方案，因为它们更好地考虑到不断变化的条件。如果考虑到这种不确定性，运营决策可能会有很大不同。然而，建模任务要复杂得多，数值计算的成本也更高。这项建议的目的是探索随机和动态规划算法的发展，并将它们应用于不同的领域。我们关注的是由情景建模的不确定性，它代表了一系列可能的结果。我们首先研究多阶段随机规划的新发展，以便利用最初在非线性规划中提出的思想来提出更快、更准确的技术。作为一个主要的应用，我们考虑了依赖于不确定水量的水力发电。更好的管理可以节省成本，扩大生产能力，还可以减少对环境的影响。其次，我们考虑了动态离散选择模型，描述了人们在有限的选择集合中的选择，但我们没有研究标准的准时模型，而是整合了人们如何在时间范围内计划决策。我们将这些思想应用到消费者行为的描述中，并在选择两点之间的道路时考虑了交通网络中做出的决策序列。所有这些问题都需要有效的估计技术，我们为底层的优化算法开发了新的策略。

Operational research now occupies an important place in research and industrial world. Along with the development of computational capacities, researchers and practitioners are more concerned about the optimization of operational processes, or the estimation of predictive models. Such models however suffer from many underlying simplifications, a major one being the assumption that all parameters are perfectly known. This is unfortunately rarely true, and a growing interest exists to incorporate uncertainty in mathematical problems. This allows to produce more flexible and robust solutions, as they better take changing conditions into account. The operational decisions can greatly differ when such an uncertainty is taken into account. However, the modeling task is then much more complex, and numerical efforts are more costly. The purpose of this proposal is to explore developments for algorithms in stochastic and dynamic programming, and to apply them to various fields. We focus on uncertainty modeled by scenarios, that represents a sequence of possible outcomes. We first study new developments in multistage stochastic programming, in order to propose faster while accurate techniques, capitalizing on the integration of ideas originally proposed in nonlinear programming. As a main application, we consider the hydroelectric generation, that is dependent of uncertain water intputs. A better management can lead to cost savings, a larger production capacity, and also a reduced impact of environment. Secondly, we consider dynamic discrete choice models, describing people choices among a finite set of alternative, but instead of studying standard punctual models, we integrate how people plan decisions over a time horizon. We apply the ideas to the description of consumers behavior, and we also consider the sequence of decisions made in a traffic network when selecting a road between two points. All these problems require efficient estimation techniques, and we develop new strategies for the underlying optimization algorithms.