Multistage Stochastic Convex Optimization

具有不确定性的优化问题在许多应用中都会遇到。一些受到相当多关注的例子包括需求不确定的库存控制、收益和负债不确定的投资组合选择，以及需求不确定的发电。建模和解决此类问题有多种方法。研究人员考虑了多阶段随机规划和动态规划问题，在这些问题中，做出了一系列决策，一些问题参数是随机的，随着时间的推移，可以在以后的决策中使用的信息变得可用。在过去的几年里，我们在处理具有不确定性的优化问题的能力方面取得了相当大的进步。从理论上证明了抽样方法的正确性，并在数值实验中得到了证实，在某些情况下甚至可以精确地解决这些问题。似乎我们现在有了良好的理论背景和一些数值经验表明，通过抽样技术可以有效地解决这些类型的问题。这项研究的目的是发展解决不确定性多阶段优化问题的基本理论和数值方法。为了使其发挥作用，必须将有效的确定性优化算法与仿真方法有效地结合起来。如果成功，它可能会为解决更多类别的现实世界问题打开可能性。在这方面，所提出的方法的理论结果和初步的数值实验是相当令人鼓舞的。

Optimization problems with uncertainty are encountered in many applications. Some examples that have received a fair amount of attention include inventory control with uncertain demand, investment portfolio selection with uncertain returns and liabilities, and electricity generation with uncertain demand. There are various approaches to modeling and solving such problems. The investigators consider Multistage Stochastic Programming and Dynamic Programming problems in which a sequence of decisions is made and some of the problem parameters are random, and information that can be used in later decisions become available over time. Considerable progress has been made in the last few years in our ability to handle optimization problems with uncertainty. It was shown theoretically and confirmed in numerical experiments that sampling methods allow solving some of these problems with proven accuracy and in some cases even exactly. It seems that we now have a sound theoretical background and some numerical experience indicating that come types of problems can be solved effectively by sampling techniques. The proposed research is aimed at developing basic theory and numerical procedures for solving multistage optimization problems with uncertainty. In order to make them work, effective deterministic optimization algorithms should be combined with simulation methods in an efficient way. If successful, it may open the possibility to solve a considerably larger class of real world problems. In that respect theoretical results and preliminary numerical experiments with the suggested approach are quite encouraging.