Mathemetical Sciences: Variational Analysis and Dynamical Optimization

[9500957]洛克菲勒这个项目主要研究与动态结构系统的管理或控制有关的优化问题。对于连续时间的问题，例如在最优控制mf工程系统和经济模型中出现的问题，将开发新形式的变分分析，以便在更具体和更广泛的设置中表征解决方案。特别地，解的稳定性问题将被研究。对于离散时间问题，不确定情况下的决策模型将成为研究的重点。将阐明随机规划和动态规划之间的联系，并为设计求解计算方法的新方法奠定基础。这项工作将涉及必要的数学创新，以理解在迄今为止无法有效建模的情况下不同控制策略的影响。特别重要的是用数学方法来表示问题结构中随机元素随时间变化的信息。为此目的，我们将探讨场景树与状态变量动力学的结合。扩展线性二次规划将被用来允许约束的惩罚表达式，并促进二元化，从而导致数值方案，其中巨大规模的问题可以迭代地分解为可在高速计算机上解决的问题。＊＊＊

9500957 Rockafeller This project centers on optimization problems concerned with the management or control of systems with dynamical structure. For problems in continuous time, such as arise in the optimal control mf engineering systems and economic models, new forms of variational analysis will be developed in order to characterize solutions more concretely and in a wider variety of settings. In particular, issues of solution stability will be studied. For problems in discrete time, models of decision-making under uncertainty will be the focus. Connections between stochastic programming and dynamic programming will be elucidated and made the basis of new approaches to the design of computational methods for finding solutions. The work will deal with mathematical innovations necessary to understand the effects of different control strategies in situations that heretofore have resisted effective modeling. Especially important are the ways of representing mathematically the revelation over time of information about random elements in a problem's structure. A combination of scenario trees with state-variable dynamics will be explored for this purpose. Extended linear-quadratic programming will be utilized to allow penalty expression of constraints and to promote dualization leading to numerical schemes in which problems of enormous scale can be broken down iteratively into problems solvable on high-speed computers. ***