Research on Stochastic Systems and Optimization: Analysis, Algorithms, and Computations

本研究项目着眼于随机优化的新算法设计和研究新兴应用中出现的系统的基本性质，包括以下四个方面。(1)发展具有马尔可夫切换的随机逼近算法，研究马尔可夫调制随机序列；它揭示了极限开关扩散、大偏差、强不变性和遍历性等渐近性质。(2)给出了连续状态依赖切换的状态切换随机微分方程的数值解。除收敛性外，第二阶段还将研究算法的收敛率、相关控制问题的数值方法及其收敛率。(3)利用马尔可夫参数和二值或量化观测值进行系统辨识。它有助于在有限的传感器信息下理解可跟踪性、复杂性和建模能力。(4)分析了状态依赖切换的切换跃扩散的稳定性，给出了非线性系统稳定与不稳定的充分条件和线性化系统的充分必要条件。通过深入的分析和广泛的数值实验，我们的目标是获得新的见解，并推动随机优化方法和随机系统理论的发展。本研究计划的动机是在无线通讯、自适应讯号处理、生产计划、排队系统、生物、生态和经济系统等新兴应用中，不可避免地涉及不确定性。与现有文献中常用的模型不同，系统也经常受到随机环境的影响。例如，当两个或两个以上的物种生活在附近，并且有相同的基本需求时，它们通常会竞争资源、食物、栖息地或领土。传统模型使用（随机或非随机）微分方程来处理这种情况。然而，系统经常受到额外的环境噪声的影响，这些噪声不能用传统的微分方程来描述。其他例子包括保险风险模型和离子通道（生物纳米管）动力学等。所提出的项目旨在考虑这种随机环境和其他不确定因素。它提出了优化任务的新算法，设计了求解方程组的数值程序，对参数未知和传感器信息有限的系统进行了识别任务，并获得了涉及连续动力学和离散事件的系统的长期行为。要研究的模型，要开发的数值算法，以及要获得的见解将共同为随机优化领域做出贡献，并对上述应用产生影响。几个研究生参与了这个研究项目。通过将所提出的研究与教学相结合，计划工作有助于进一步发展随机优化和随机系统理论，提高数学教育水平。

Focusing on new algorithms design for stochastic optimization and investigating basic properties of systems arising in emerging applications, this research project encompasses the following four aspects. (1) It aims to develop stochastic approximation algorithms with Markovian switching and examining Markov modulated random sequences; it reveals asymptotic properties such as limit switching diffusions, large deviations, strong invariance, and ergodicity. (2) It presents numerical methods for solutions of regime-switching stochastic differential equations with continuous-state-depend switching. In addition to convergence, in the second phase, rates of convergence of the algorithms, numerical methods for the related control problems, and their convergence rates are to be investigated. (3) It carries out system identification with Markov parameter and binary-valued or quantized observations. It facilitates the understanding of tractability, complexity, and modeling capability under limited sensor information. (4) It analyzes stability of switching jump diffusions with state-dependent switching, and provides sufficient conditions for stability and instability of nonlinear systems and necessary and sufficient conditions for linearizable systems. Consisting of in-depth analysis and extensive numerical experiments, our goals are to gain new insight, and to advance state of the art of stochastic optimization methods and stochastic systems theory.This research project is motivated by emerging applications arising in wireless communication, adaptive signal processing, production planning, queueing systems, biological, ecological, and economic systems, which are inevitably involve uncertainty. In contrast to the usual models in the existing literature, the systems are often influenced by random environment as well. For example, when two or more species live in proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. Traditional models use (either random or non-random) differential equations for such scenarios. However, the systems are often subject to additional environmental noise, which cannot be described by the traditional differential equation setup. Other examples include insurance risk models and ion channel (biological nanotubes) dynamics among others. The proposed project aims to take into such random environment and other uncertain factors into consideration. It presents novel algorithms for optimization tasks, designs numerical procedures for solving systems of equations, carries out identification task for systems with unknown parameters and limited sensor information, and obtains longtime behavior of systems involving both continuous dynamics and discrete events. The models to be examined, the numerical algorithms to be developed, and the insight to be gained will jointly contribute to the field of stochastic optimization and make impact on the aforementioned applications. Several graduate students are involved in the research project. By integrating the proposed research with teaching, the planned work contributes to the further development of stochastic optimization and stochastic systems theory and the improvement of mathematics education.