Mathematical Sciences: Analysis and Numerical Methods in Stochastic Optimization

小行星9529738 这个随机优化的项目包括四个部分：一个考试 奇异扰动马尔可夫链的性质;全局随机优化算法的发展;被动随机逼近过程的研究;以及制造系统随机优化的数值方法设计。 第一部分主要研究了奇摄动非平稳马氏链和具有弱相互作用和强相互作用的奇摄动马氏链的渐近性质，并得到了相关控制问题的渐近最优性。 第二部分包括全局随机优化算法的渐近性质的发展。 通过使用弱收敛方法，收敛性和收敛速度是要建立的，并适当的缩放序列的适当的随机微分方程的解的收敛性是要确定的。 第三部分将随机逼近方法与核估计相结合，得到被动随机逼近算法。 这里的目标是建立收敛的算法，并获得误差估计。 第四部分包括解决一类分段确定性马尔可夫链的鲁棒控制问题的数值方法，以及多机制造系统在半成品策略下近似最优阈值的随机梯度下降算法的发展。 得到了算法的收敛性和渐近性。 整个研究项目将包括两个部分：分析和模拟。主要目标是加强基本的了解的渐近性质的基本系统，并制定健全和可行的算法。 该提案包括四个密切相关的专题。这是随机优化理论与应用之间的桥梁。 强调的是 通过分析和数值实验开发有效的数值方法。第一个主题涉及的系统是受快速和随机变化。所获得的结果是有用的应用，涉及层次决策，生产计划，排队网络的通信，和系统的可靠性。为了满足日益增长的高效需求， 数值程序的全局优化，该项目的第二部分侧重于数值算法的发展时，随机误差的数据必须考虑在内。 结果是适用于不断扩大的应用范围的估计，识别和优化问题。 在本计划的第三部分中进行的研究来自化学工程中的稳态估计和检测问题（连续搅拌槽和二元蒸馏塔）。 重点是数值算法的设计和实现，预计这些算法在目标识别，跟踪，系统故障检测，信号处理和相关领域也有各种应用。该计划的第四部分是关于不可靠的制造系统的优化方法。 通过强调系统的稳定性而不仅仅是最优性，发展了一类生产计划鲁棒控制问题的数值方法。 根据目前流行的看板政策，这最初是由日本汽车工业，一个数值程序，将提供一个系统的方法来寻找制造系统的看板的最佳数量是开发。 ***

9529738 Yin This project in stochastic optimization consists of four parts: an examination of the properties of singularly perturbed Markov chains; a development of global stochastic optimization algorithms; a study of passive stochastic approximation procedures; and the design of numerical methods for stochastic optimization in manufacturing systems. The first part is intended to derive asymptotic properties of singularly perturbed nonstationary Markov chains and perturbed chains with weak and strong interactions, and to obtain asymptotic optimality in related control problems. The second part comprises the development of asymptotic properties of global stochastic optimization algorithms. By using weak convergence methods, convergence and rates of convergence are to be established, and the convergence of suitably scaled sequences to the solutions of appropriate stochastic differential equations is to be ascertained. In the third part, by combining stochastic approximation methods with kernel estimators, passive stochastic approximation algorithms are to be obtained. Here the goal is to establish convergence of the algorithms and to obtain error estimates. The fourth part encompasses numerical procedures for solving a class of robust control problems of piecewise deterministic Markov chains, and the development of stochastic gradient descent algorithms for approximating the optimal threshold values under the WIP (work in progress) policies for multi-machine manufacturing systems. Convergence and asymptotic properties of the algorithms are to be obtained. The entire research project will include two components: analysis and simulation. The main objectives are to enhance basic understanding of the asymptotic properties of the underlying systems and to develop sound and feasible algorithms. This proposal consists of four closely related topics. It is a bridge-building attempt to link theory and application in stochastic optimization. Emphasis is on the development of efficient numerical methods via analysis and numerical experimentation. The first topic deals with systems that are subject to rapid and random variations. The results obtained are intended be useful for applications involving hierarchical decision making, production planning, queuing networks in communication, and system reliability. To meet the increasing demand for efficient numerical procedures for global optimization, the second part of the project focuses on the development of numerical algorithms when random errors in the data have to be taken into account. The results are applicable to an ever expanding range of applications in estimation, identification and optimization problems. The research undertaken in the third part of this program derives from steady state estimation and detection problems in chemical engineering (for a continuously stirred tank and for a binary distillation column). The emphasis is on the design and implementation of numerical algorithms, which are expected to also have a variety of applications in target recognition, tracking, system failure detection, signal processing and related fields. The fourth part of the program is concerned with optimization methods for unreliable manufacturing systems. By emphasizing system stability rather than optimality alone, numerical methods for a class of robust control problems of production planning are to be developed. Under nowadays popular Kanban policies, which were originally promoted by the Japanese auto industry, a numerical procedure that will provide a systematic way of finding the optimal number of Kanbans for manufacturing systems is to be developed.. ***