Behavioural-based mathematical programming: algorithmics and applications

我的研究目的是从理论和计算的角度研究涉及目标相互冲突的主体的决策问题。为了修正想法，让我们考虑一下高速公路连接处的收费问题。为了优化收入，必须考虑司机的行为，他们希望将自己的出行成本降至最低。如果通行费低，收入就会低。如果通行费高，收入也会低，因为高速公路对司机的数量很少，对他们来说很有吸引力。然后，人们必须在考虑到用户行为的情况下取得适当的平衡。尽管已经表明，即使在这种简单的情况下，问题也很难解决，但当整个人口的行为不同或发生拥堵时，情况就会变得更加复杂。这样的问题符合双层数学规划的框架，也就是领导者-追随者游戏。在这些博弈中，领导者将对手或受其决策影响的群体的理性反应纳入其优化问题。在我们的介绍性示例中，这种反应将是司机的路径选择。这种情况无处不在。事实上，一个问题的所有变量都由领导者控制，这是一个例外。*双层规划已应用于能源(优化网络效率、监控“智能电网”)、城市规划、医疗保健、交通(铁路、航空)等多个领域。各国政府利用监管和税收来增加全球福利，如果他们不考虑公民面临的替代方案，就无法实现这一目标。例如，在卷烟大幅增税引发走私之后，导致税收减少和卷烟消费实际增加，这与所追求的目标背道而驰！*尽管双层模型能够解决重要的工业、商业和社会问题，但它们的计算复杂性受到限制。因此，设计利用每个应用程序的结构的智能算法非常重要。对于我正在研究的一类问题，一种方法是将原始模型近似为更容易处理的模型，然后从其解向原始模型的解取得进展。基于双层问题的组合性质的其他策略也将被研究。所有这些都将应用于能源、设施选址(包括医疗保健)和交通运输等领域的实际情况。

The aim of my research is to study, from both the theoretical and computational points of view, decision problems that involve agents with conflicting objectives. To fix ideas, let us consider the problem of setting tolls on the links of a highway. In order to optimize the revenue, one must take into account the behaviour of drivers, who want to minimize their own travel cost. If tolls are low, revenue will be low. If tolls are high, revenue will also be low, due to the low number of drivers for which the highway will be attractive. One must then strike the right balance, taking into account user behaviour. Although it has been shown that, even in that simple setting, the problem is hard to solve, the situation becomes all the more complex when behaviour varies across the population or when congestion occurs. Such problems fit the framework of bilevel mathematical programs, or leader-follower games. In these games, the leader factors into its optimization problem the rational reaction of an adversary or a population impacted by his decisions. This reaction would be the drivers' path choices in our introductory example. This situation is ubiquitous. Indeed, it is the exception that all variables of a problem are controlled by the leader. ******Bilevel programming has been applied to areas as diverse as energy (optimizing the efficiency of a network, monitoring the 'smart grid'), urban planning, health care, transportation (rail, airlines). Governments use regulations and taxes to increase global welfare, and fail to achieve this goal if they do not consider the alternatives faced by the citizens. For instance, following a stiff tax increase of cigarettes that triggered smuggling, resulting in a decrease of tax revenues and an actual increase in cigarette consumption, contrary to the goal pursued!******Despite their capability to address important industrial, commercial and social issues, bilevel models are limited by their computational complexity. It is therefore important to design smart algorithms that take advantage of each application's structure. For a class of problems that I am studying, one approach consists in approximating the original model by a more tractable one, and then make progress from its solution towards the solution of the original. Other strategies, that are based on the combinatorial nature of bilevel problems, will also be investigated. All of them will be applied to practical instances in energy, facility location (including health care) and transportation.