On the exploitation of uncertainty in exact and approximate optimization

随着计算能力的进步，涉及不确定性的数学程序越来越多地被研究，因为它们往往更好地捕捉问题的不完整知识进行优化，特别是当它涉及到未来的决策时。本研究旨在分析如何处理一些实际问题中的不确定性，并改进不确定性的开发。******第一个目标是在飞机到达顺序的背景下考虑不确定性，因为计划范围预计将在未来几年内增长，而有效的到达顺序允许增加机场跑道的使用，同时满足运营限制，特别是在安全考虑方面。更具体地说，调度必须提前执行，以限制飞机到达机场时的空中管制操作。******第二个更一般的目标是考虑涉及一系列决策的问题，分阶段进行。第一阶段的决定是最重要的，因为它们必须首先执行，但是我们必须考虑到随后的阶段，以便选择不会在未来造成问题的第一阶段行动。即使只考虑两个阶段，这类问题的复杂性也会迅速增加，特别是如果要解决的第二阶段程序很复杂，可能涉及黑盒优化或模拟。我们的目的是研究在实践中需要解决多少个对应于不同场景的第二阶段问题才能获得足够质量的第一阶段解，以及在第二阶段问题只能近似解决的情况下，建立第二阶段方案解所需的精度。******我们还计划将这些发现应用于动态离散选择的背景下，其中给定的个体必须在一组离散的备选方案之间操作一系列选择，可能与时间相关。一个具体的应用是路径选择估计问题，因为最近的进展允许通过动态规划有效地解决它，只要网络是完全已知的决策者在原点。已经证明，将问题表示为一系列环节的选择提供了一个更易于处理的表述。即使完美知识假设是限制性的，它也可以利用最短路径和动态规划之间的类比，在确定性设置中估计离散选择序列。对随机情况的扩展不是微不足道的，但我们的目标是利用近似动态规划的发展来解决它们。******另一个次要目标是分析随机噪声如何直接用于优化以使搜索多样化，因为它可以通过在导数自由优化和元启发式中提出的混合随机搜索技术帮助非线性问题逃避局部最小化。

With the progress in computing power, mathematical programs involving uncertainty are more and more studied as they often better capture the incomplete knowledge of the problem to optimize, especially when it involves future decisions. The proposed research aims to analyze how to handle uncertainty on some practical problems and to improve its exploitation.******A first objective is to take uncertainty into consideration in the context of aircraft arrival sequencing, as the planning horizon is expected to grow during the next years, while an efficient arrival sequencing allows to increase the use of the airport runways while satisfying the operational constraints, especially with respect to safety considerations. More specifically, the scheduling has to be performed in advance in order to limit the air control operations when the aircraft arrives at the airport.******A second, more general, objective is to consider problems involving a sequence of decisions, over stages. The first-stage decisions are the most important, as they have to be implemented first, however we have to take account of the subsequent stages in order to select first-stage actions that will not cause problems in the future. The complexity of such problems increases very fast, even when only two stages are considered, especially if the second-stage programs to solve are complex, possibly involving black-box optimization or simulation. We aim to study how many second-stage problems, corresponding to different scenarios, should be solved in practice to obtain a first-stage solution of sufficient quality, and in the case where the second-stage problem can only be solved approximately, to establish the required accuracy for the second-stage program solution.******We also project to apply the findings in the context of dynamic discrete choice, where a given individual has to operate a sequence of choices between a discrete set of alternatives, possibly time-dependent. A specific application is the route choice estimation problem as recent progress allows to efficiently solve it by means of dynamic programming, as long as the network is perfectly known by the decision-maker at the origin. It has been shown that representing the problem as the choice of a sequence of links delivers a more tractable formulation. Even if the perfect knowledge assumption is restrictive, it opens the possibility to estimate discrete choice sequences in a deterministic setting, using the analogy between shortest path and dynamic programming. The extension to the stochastic situation is not trivial but we aim to capitalize on developments in approximate dynamic programming to address them.******A side objective is also to analyze how random noise can be directly exploited in optimization to diversify the search, as it can help to escape local minimizers in nonlinear problems, by hybridizing random search techniques proposed in derivative free optimization and metaheuristics.