Mathemetical Sciences: Variational Analysis and Dynamical Optimization

9500957 Rockafeller该项目的重点是与动态结构系统的管理或控制相关的优化问题。 对于连续时间的问题，如工程系统和经济模型中的最优控制问题，将发展新形式的变分分析，以便更具体地描述解决方案的特征，并在更广泛的设置中。 特别是，将研究溶液稳定性问题。 对于离散时间问题，不确定性条件下的决策模型将是研究的重点。 随机规划和动态规划之间的联系将得到阐明，并提出了新的方法来设计计算方法的基础上找到解决方案。 这项工作将涉及必要的数学创新，以了解不同的控制策略在迄今为止一直抵制有效建模的情况下的影响。 尤其重要的是，如何用数学的方法来表示问题结构中随机元素的信息随时间的变化。 为此目的，将探讨与状态变量动态的情景树的组合。 扩展的线性二次规划将被用来允许惩罚表达式的约束，并促进二元化导致的数值方案，其中巨大规模的问题可以分解成可解决的问题，在高速计算机上迭代。 ***

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. ***