Towards new solution techniques in mathematical programming with scenarios

运筹学在研究和工业界占有重要的地位。沿着计算能力的发展，研究者和实践者更加关注操作过程的优化，或者预测模型的估计。然而，这些模型受到许多潜在的简化，其中一个主要的假设是所有参数都是完全已知的。不幸的是，这很少是真的，并且越来越多的人对将不确定性纳入数学问题中感兴趣。这可以产生更灵活和更强大的解决方案，因为它们可以更好地考虑不断变化的条件。当考虑到这种不确定性时，业务决策可能会大不相同。然而，建模任务要复杂得多，数值工作的成本也更高。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。我们专注于通过场景建模的不确定性，这些场景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，以提出更快而准确的技术，利用最初在非线性规划中提出的想法的整合。作为一个主要的应用，我们考虑水力发电，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，并减少对环境的影响。其次，我们考虑动态离散选择模型，描述人们的选择之间的一组有限的替代品，但不是研究标准的准时模型，我们整合人们如何计划在一个时间范围内的决策。我们应用的想法来描述消费者的行为，我们也考虑了在交通网络中选择两点之间的道路时所做的决定的顺序。所有这些问题都需要有效的估计技术，我们开发的基础优化算法的新策略。

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.