Variational Analysis in Problems of Optimization

Pang，Jong-Shi发件人：Terry Rockafellar [rtr@math.washington.edu]发送人：2001年6月13日，星期三下午3点59分收件人：jpang@nsf. gov主题：提案摘要钟世，这是我的提案“最优化问题中的变分分析”的摘要。 特里 ~~ 将研究参数依赖中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和从非标准的角度。 对于某些类型的最优控制，成本去功能和反馈之间的联系将被探索沿着最近推广的欧拉-拉格朗日方程和哈密顿方程的控制框架的专业化。 新的风险模型和拆分算法将在随机规划中开发，其中必须在充分信息之前做出最佳决策。 这些努力是必要的，因为优化问题，这是非常重要的许多应用程序，从管理到工程，没有很好地涵盖了过去的标准数学，并要求新的思维方式。 例如，在以最小化“成本”或最大化“效率”为中心的此类问题中，通常需要满足非常多的副条件，而且需要处理的决策变量数量可能非常巨大--数千甚至数百万。 至关重要的是要有一个坚实的基础，为发展的数值方法，找到解决方案，并了解输入数据的变化可能对这些解决方案的影响。 有待开展的工作将推动这些方面的进展。

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.