Variational Analysis in Problems of Optimization

Pang，Jong-Shim来源：Terry Rockafella[RTR@math.washington.edu]发送时间：2001年6月13日星期三3：59 PMTO：jpang@nsf.Government主题：提案摘要Jong-Shih，这是我提案《优化问题中的变分分析》的摘要。Terry~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~这个项目将把变分分析的最新进展应用于一系列优化问题和问题。将研究参数相依的灵敏度和度量正则性。二阶最优性条件将被设计用于新的目的，并从非标准角度出发。我们将探索某些类型的最优控制的成本函数和反馈之间的联系，以及最近推广的欧拉-拉格朗日方程和哈密顿方程的控制框架的专门化。在随机规划中将开发新的风险模型和拆分算法，其中必须在完全信息之前做出最优决策。这些努力是必要的，因为优化问题在从管理到工程的许多应用中都非常重要，过去的标准数学不能很好地涵盖这些问题，需要新的思维方式。例如，在以最小化“成本”或最大化“效率”为中心的这类问题中，通常有非常多的附带条件需要满足，而且需要处理的决策变量的数量可能是巨大的-数千甚至数百万。至关重要的是要有一个坚实的基础，以便开发寻找解决方案的数值方法，并了解输入数据的变化可能对这些解决方案产生的影响。将要开展的工作将推动在这些方向上取得进展。

Pang, Jong-ShiFrom: Terry Rockafellar [rtr@math.washington.edu]Sent: Wednesday, June 13, 2001 3:59 PMTo: jpang@nsf.govSubject: proposal abstract Jong-Shih, here's the abstract for my proposal "Variational Analysis in Problems of Optimization". Terry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This project will apply the latest advances in variational analysis to a range of optimization problems and issues. Sensitivity and metric regularity in parametric dependence will be studied. Second-order optimality conditions will be devised for new purposes and from nonstandard perspectives. Connections between cost-to-go functions and feedback for certain kinds of optimal control will be explored along with the specialization to a control framework of recently generalized Euler-Lagrange equations and Hamiltonian equations. New risk models and splitting algorithms will be developed in stochastic programming, where decisions must be made optimally in advance of full information. These efforts are needed because problems of optimization, which are of great importance in many applications from management to engineering, are not well covered by standard mathematics of the past and have required new ways of thinking. In such problems, centered on minimizing "cost" or maximizing "efficiency", for instance, there usually are very many side conditions that have to be satisfied, and moreover the number of decision variables to be coped with can be enormous---in the thousands or even millions. It is crucial to have a solid foundation for the development of numerical methods for finding solutions and understanding the effects that shifts in input data might have on those solutions. The work to be performed will further the progress in those directions.