Novel Algorithms to Approximate the Future Consequence of Sequential Decisions

在商业、医疗保健和交通运输中出现的许多复杂问题可以建模为不确定条件下的顺序决策问题，这意味着决策者必须在一些随机事件随时间展开时周期性地做出决策。例如，一家航空公司在不知道未来实际需求的情况下，动态地改变城市网络中不同航班的票价，试图在管理未售出座位风险的同时最大化其收入。这些问题可以方便地以动态规划的形式进行建模，动态规划是一种通过最大化当前奖励和预期未来奖励的总和来寻找最佳决策的方法。不幸的是，对于许多实际问题，为了计算期望的未来奖励函数，人们应该考虑的未来场景的数量是指数级的，这使得该函数的精确计算变得棘手。为了克服这一问题，提出了近似动态规划（ADP）方法来寻找近似最优解。

Many complex problems arising in business, health care, and transportation can be modelled as sequential decision making problems under uncertainty, meaning that a decision maker has to make decisions periodically while some random events unfold over time. For instance, an airline dynamically changes the fare for different flights over a network of cities without knowing the actual future demand, trying to maximize its revenue while managing the risk of unsold seats. These problems can be conveniently modelled in the form of dynamic programs, a method that finds the best decision by maximizing the sum of immediate reward and the expected future reward. Unfortunately, for many practical problems, the number of future scenarios that one should consider in order to calculate the expected future reward function is exponentially large, making exact calculation of this function intractable. In order to overcome this issue, approximate dynamic programming (ADP) methods have been developed to find an approximate optimal solution. A cornerstone of many ADP algorithms is defining a set of basis functions (or an approximation architecture) for approximating the future consequence of present decisions (the expected future reward function). Currently, the choice of basis functions requires prior expert knowledge about the problem, and is usually considered as more of an art than a science. My research program aims to develop, study, and apply novel algorithms that automate generation of basis functions by efficiently selecting a subset of functions from a large pool of potential basis functions, and updating this set as more information becomes known about the problem. The benefit of such algorithms is twofold: first, it reduces the burden to come up with a well-informed set of basis functions that requires significant prior knowledge about the problem; second, since many potential candidates are considered for basis functions, it is expected that the quality of the approximation is improved. My short-term objective includes evaluating the performance of the proposed algorithms in a variety of application areas, such as perishable inventory management, patient scheduling, and revenue management. ADP is a general method that is commonly used for solving many different problems in a variety of applications. As the quality of the policies generated by these algorithms is dependent on the quality of the basis functions chosen, it would be of great interest, both theoretically and practically, if the process of generating and selecting basis functions can be automated. Therefore, even a small improvement achieved by the findings of my research would have significant practical implications in multiple application areas.