Mathematical Sciences: Second-Order Optimality Conditions for Problems with Constraints

数学科学：带约束问题的二阶最优性条件

• 批准号: 9404591
• 负责人: Vera Zeidan
• 金额: $ 4万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-11-01 至 1996-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404591&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Second; Order; Optimality

9404591 Zeidan A general optimal control problem is considered in which the control and/or state constraints are of the form of an inclusion in a convex set. The accessory problem when the data functions are differentiable with Lipschitz derivatives will be derived. A new term is expected to appear and hence, this is a generalization of the envelope- like effect (introduced by Kawasaki). The results will be expressed only in terms of the original data. New second- order necessary conditions for strong local minimality of the above problem form the second part of this project. This is a long-standing problem. The key idea of this research is based on suitably modifying the time transformation, introduced by A. Ya. Dubovitskii and A. A. Milyutin, in order to obtain a new problem to which the second-order necessary conditions for weak local optimality (developed in the first part of the proposal) could be applied. The third part of this project is to develop the sufficiency theory for the case when pure state inequality constraints and control set constraints are present. This research involves a modified Riccati equation and the study of connections between the three concepts: Riccati equation, coupled points, and positive definiteness of the second variations. This project lies at the heart of optimal control theory, a subject which was developed in the 1950's mainly to deal with problems emanating from engineering, natural resources, and economics. This field, which is a branch of optimization, could be described as follows: Find a candidate that produces the smallest cost (or the largest return) while meeting given requirements. The latter are referred to as constraints. "First order optimality conditions'' serve to produce a list of possible candidates for the smallest cost. In practice, many of them do not furnish the least cost. Hence, there is a need for another test to screen the candidates and to obtain a short list. This test evok es what are known as "second order conditions''. In this project, second-order optimality conditions for optimal control problem with very general constraints will be derived. In fact, several recent problems in aerospace science, industrial process control, robotics and mathematical economics can be formulated as optimal control problems with such general constraints (state and/or control set). However, in this complicated setting no satisfactory second order conditions are presently available. Thus, when accomplished, this research will provide new tools to assist engineers and economists in the selection of optimal strategies. ***

小行星9404591 一般的最优控制问题被认为是控制和/或状态约束的形式包含在一个凸集。 当数据函数与Lipschitz导数可微时，将导出辅助问题。 一个新的术语预计将出现，因此，这是一个概括的信封样效应（由川崎介绍）。结果将仅以原始数据表示。 本文的第二部分给出了上述问题强局部极小的新的二阶必要条件。这是一个长期存在的问题。本研究的核心思想是对A。Ya. Dubovitskii和A. A.为了得到一个新的问题，弱局部最优性的二阶必要条件（在提案的第一部分中提出）可以应用。 该项目的第三部分是针对存在纯状态不等式约束和控制集约束的情况发展充分性理论。 本研究涉及一个修正的Riccati方程，并研究了Riccati方程、耦合点和二次变分的正定性这三个概念之间的联系。 这个项目是最优控制理论的核心，最优控制理论是在20世纪50年代发展起来的，主要用于处理工程、自然资源和经济学方面的问题。这个字段是优化的一个分支，可以描述如下：找到一个在满足给定要求的同时产生最小成本（或最大回报）的候选项。 后者被称为约束。 “一阶最优性条件”用于产生最小成本的可能候选者列表。 在实践中，他们中的许多人并没有提供最低的成本。 因此，需要进行另一次测试来筛选候选人并获得短名单。 这个测试唤起了所谓的“二阶条件”。 在这个项目中，二阶最优性条件的最优控制问题的非常一般的约束将被导出。 事实上，最近在航空航天科学，工业过程控制，机器人和数学经济学中的一些问题可以用公式表示为具有这种一般约束（状态和/或控制集）的最优控制问题。 然而，在这种复杂的设置没有令人满意的二阶条件目前可用。 因此，完成后，这项研究将提供新的工具，以协助工程师和经济学家在选择最佳的战略。 ***