Towards new solution techniques in mathematical programming with scenarios

运筹学目前在研究界和工业界占有重要地位。随着计算能力的发展，研究人员和实践者越来越关注操作流程的优化，或预测模型的估计。然而，这样的模型遭受了许多潜在的简化，一个主要的是假设所有的参数都是完全已知的。不幸的是，这很少是正确的，人们对将不确定性纳入数学问题的兴趣越来越大。这允许产生更灵活和健壮的解决方案，因为它们更好地考虑到不断变化的条件。当考虑到这种不确定性时，操作决策可能会有很大的不同。然而，这样的建模任务就复杂得多，数值计算的成本也更高。本提案的目的是探索随机和动态规划算法的发展，并将其应用于各个领域。我们关注的是情景模拟的不确定性，它代表了一系列可能的结果。我们首先研究了多阶段随机规划的新发展，以便提出更快而准确的技术，利用非线性规划中最初提出的思想的整合。作为主要应用，我们考虑了水力发电，这是依赖于不确定的水输入。更好的管理可以节省成本，提高生产能力，减少对环境的影响。其次，我们考虑动态离散选择模型，描述人们在有限的备选集中的选择，但不是研究标准准时模型，我们整合了人们如何在一个时间范围内计划决策。我们将这些想法应用于消费者行为的描述，并且我们还考虑了在两点之间选择道路时在交通网络中做出的决策顺序。所有这些问题都需要有效的估计技术，我们为底层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.