Behavioural-based mathematical programming: algorithmics and applications

我的研究目的是从理论和计算的角度来研究涉及具有冲突目标的代理的决策问题。为了解决这个问题，让我们考虑一下在高速公路的连接处设置通行费的问题。为了优化收益，必须考虑司机的行为，他们想要最小化自己的出行成本。如果通行费很低，收入就会很低。如果收费很高，收入也会很低，因为高速公路对司机的吸引力很小。开发者必须在考虑用户行为的情况下取得适当的平衡。尽管研究表明，即使在这种简单的情况下，这个问题也很难解决，但当人群的行为发生变化或出现拥堵时，情况就变得更加复杂了。这样的问题符合双层数学程序的框架，或者说是领导者-追随者的游戏。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，这种反应就是驾驶员的路径选择。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是例外。******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）等不同领域。政府利用法规和税收来增加全球福利，如果不考虑公民面临的其他选择，就无法实现这一目标。例如，在对卷烟大幅增税后，引发了走私，导致税收减少，实际卷烟消费量增加，这与所追求的目标背道而驰！******尽管它们有能力解决重要的工业、商业和社会问题，但双层模型受到其计算复杂性的限制。因此，设计利用每个应用程序结构的智能算法是很重要的。对于我正在研究的一类问题，一种方法是用一个更容易处理的模型逼近原始模型，然后从它的解向原始模型的解推进。其他基于双层问题的组合性质的策略也将被研究。所有这些都将应用于能源、设施选址（包括医疗保健）和运输方面的实际情况。

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.