Optimization for Systems Under Uncertainty: Modeling, Asymptotic Analysis, and Recursive Algorithms

提案标题：不确定性下的系统优化：建模、渐近分析和递归算法提案编号：dms-9877090PI：G.George YinAff.：密歇根州底特律韦恩州立大学数学系，密歇根州48202。313-577-2496，传真：313-577-7596，电子邮件：gyin@math.wayne.eduAbstract技术描述：本提案重点关注不确定性下系统的建模和优化，该提案由四个部分组成。第一部分提出了两种算法。第一个是模拟扩散机的近似值；第二个也考虑了测量误差。我们的目标是发展这类算法的渐近性质。通过使用弱收敛方法，适当缩放的序列将被证明收敛到适当的扩散。第二部分讨论一类混合模型。对于含有弱相互作用和强相互作用的奇摄动马氏链的系统，我们将建立逼近格式，这对于大规模系统的自然时间尺度分离和降低复杂性是有用的。第三部分研究了由零递归扩散引起的柯西问题解的渐近性质。我们的重点是获得了解的收敛和收敛速度。其中一个主要动机来自对奇异摄动系统的研究。研究结果对优化、受控马尔可夫系统、递阶决策、生产计划、电信、排队网络和系统可靠性等领域的应用具有重要意义。第四部分研究了加工时间随机、机器故障和维修、加工时间随机压缩下的单机调度问题。我们的目标是建立可行的模型，为底层系统获得最优调度策略。这些结果将使人们可以通过将集成过程看作单机系统来设计更复杂的车间调度模型和策略。非技术解释：为了弥合理论和应用之间的差距，本研究项目包括建模、渐近分析和仿真三个部分。最终目的是提供有用的模型，研究它们的基本性质，并开发合理可行的算法。第一部分提出了两类算法，并将其应用于机器学习、图像分割和各种全局优化任务。为了满足特殊模式识别、信号处理、电信和制造等领域对系统健壮性设计和控制的日益增长的需求，第二部分的目的是用一个简单的系统通过近似模式来降低复杂结构的大系统的复杂性。第三部分的计划工作源于对随机影响的不确定性进行建模的努力，如制造系统中的产品需求或股票市场的波动。要控制底层系统，必须了解系统的长期行为，这是我们的首要目标。在生产计划中，为机器要加工的零件排序提供良好的策略是至关重要的。第四部分提出了不确定环境下的单机调度模型，所提出的工作旨在制定最优调度策略，总体规划工作是PI最近在这些领域初步探索的继续。期望研究结果能应用于优化方法的进一步改进。

Proposal Title: Optimization for Systems under Uncertainty: Modeling, Asymptotic Analysis, and Recursive AlgorithmsProposal Number: DMS-9877090PI: G. George YinAffl.: Department of Mathematics, Wayne State University, Detroit, MI 48202 Tel. 313-577-2496, Fax 313-577-7596, Email: gyin@math.wayne.eduAbstractTechnical Description:Focusing on modeling and optimization for systems under uncertainty, thisproposal consists of four parts. Part I proposes two types of algorithms. Thefirst one is an approximation of an analog diffusion machine; the secondone also takes measurement errors into consideration. Our goal is to developasymptotic properties of such algorithms. By using weak convergence methods,suitably scaled sequences will be shown to converge to appropriate diffusions.Part II treats a class of hybrid models. Approximation schemes forsystems involving singularly perturbed Markov chains with weak and stronginteractions will be developed, which are useful for natural time-scaleseparation and reduction of complexity for large-scale systems.Part III investigates asymptotic properties of solutions of Cauchy problemsarising from null-recurrent diffusions. Our focus is on obtaining convergenceand rate of convergence of the solutions. One of the primary motivations comesfrom the investigation of singularly perturbed systems. The results will beuseful to the ever expanding applications in optimization, controlled Markovsystems, hierarchical decision making, production planning, telecommunication,queueing networks, and system reliability.Part IV models single-machine scheduling problems under random processing time, and/or under random machine breakdownsand repairs, and/or subject to random compression of processing times.Our objectives are to develop feasible models and to obtain optimalscheduling policies for the underlying systems. These results will allow usto design scheduling models and strategies for more complex jobshops byconsidering integrated processes as single-machine systems.Nontechnical explanation:To bridge the gap between theory and applications, this research projectincludes three components: modeling, asymptotic analysis, and simulation.The ultimate goals are to provide useful models, to investigate their basicproperties, and to develop sound and feasible algorithms.Part 1 proposes two classes of algorithms with applications to machinelearning,image segmentation, and various global optimization tasks.To meet the increasing demand on robust design and control of systems inspeechand pattern recognition, signal processing, telecommunications, andmanufacturing, Part 2 aims to reduce the complexity of a large-scalesystem of complex structure by using a simple system via approximationschemes.The origin of the planned work for Part 3 stems from the effort of modelinguncertainties due to random influence such as demands for a product in amanufacturing system or fluctuation in the stock market. To controlthe underlying system, it is imperative to understand the system's long-termbehavior, which is our primary goal.In production planning, it is vital to provide good strategyin sequencing the parts to be processed by the machines. Part 4 proposessingle-machine scheduling models in uncertain environment.The proposed work aims to develop optimal scheduling policies.The overall planned work represents a continuation of the PI's recent preliminary exploration in these areas. It is expected that the results will be applicable in the further improvements of optimization methods.