Decentralized optimization and algorithms for stochastic dynamical systems with applications

随机动力系统的分散优化和算法及其应用

• 批准号: RGPIN-2014-03827
• 负责人: Huang, Minyi
• 金额: $ 1.68万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610669
• 关键词: Decentralized; optimization; algorithms; stochastic; dynamical

Large-population stochastic systems with mean field interactions arise in a broad range of backgrounds including social-economic systems, engineering (such as wireless networks, traffic systems, etc.), biological systems. In the past decade mean field game theory has experienced rapid development based on ideas in statistical physics and provides powerful tools to deal with the curse of dimensionality in dynamic competitive decision problems with many agents. This area continues to attract the attention of many researchers worldwide, discovering new theoretical results and opening up new areas of applications. Within the mean field game setup, this research program aims to develop significant applications to stochastic economic growth theory. The related classic literature is the endogenous stochastic growth models introduced by Brock and Mirman (1972) for the discrete time case, and by Merton (1975) for the continuous time case; these works form the foundation of stochastic growth theory, a long active area in economics. This research will mainly adopt the continuous time modeling while also considering certain aspects of the discrete time case. We will first generalize Merton's capital growth dynamics, described by a stochastic differential equation, to an "interacting particle system" situation and formulate a mean field game. This generalized system is used to describe the competitive behavior of a large number of economic agents engaged in a certain type of production activity. We are particularly interested in addressing the congestion effect, or called negative externality, where the increase of the aggregate capital level decreases the production efficiency of individual agents. Following the basic idea of mean field games, we will design the strategy of individuals using its own operational information and some predictable macroscopic quantity generated by the whole population, and will further examine the formation of the mean field resulting from the microscopic optimizing behavior of individuals. The mathematical machinery to be deployed to carry out this project includes optimal control theory, partial differential equations, stochastic processes, among others. The methodology and results will be of interest to applied mathematicians, economists, and system and control theorists. Another part of this research program will study social opinion dynamics in a probabilistic setting. This area is of great interest to social science, economics and statistical physicists. Our main interest is to address (i) random uncertainties which occur during a given agent's acquisition of others' opinions and (ii) free will induced noise, that is, a person's opinion may have random shift possibly due to human phycology. We will devise cautious opinion learning algorithms in such noisy environments, and study the formation of collective patterns resulting from simple opinion updating rules at the individual level. This research will offer new insights for understanding the pattern formation of certain social and cultural phenomena via mathematical modeling and analysis.

具有平均场相互作用的大种群随机系统出现在广泛的背景中，包括社会经济系统，工程（如无线网络，交通系统等），生物系统。在过去的十年中，平均场博弈理论经历了快速的发展，基于统计物理的思想，并提供了强大的工具来处理维数灾难的动态竞争决策问题的多代理。该领域不断吸引全球众多研究人员的关注，发现新的理论成果，开辟新的应用领域。 在平均场博弈的设置，本研究计划的目的是发展显着的应用随机经济增长理论。与之相关的经典文献是Brock和Mirman（1972）针对离散时间情形引入的内生随机增长模型和Merton（1975）针对连续时间情形引入的内生随机增长模型，这些工作构成了经济学长期活跃的随机增长理论的基础。本研究将主要采用连续时间建模，同时也考虑离散时间情况的某些方面。我们将首先概括默顿的资本增长动力学，描述了一个随机微分方程，一个“相互作用的粒子系统”的情况下，制定一个平均场游戏。这个广义系统用来描述从事某种类型生产活动的大量经济主体的竞争行为。我们特别感兴趣的是解决拥塞效应，或称为负外部性，其中总资本水平的增加降低了个体代理的生产效率。遵循平均场博弈的基本思想，我们将利用个体自身的操作信息和整个种群产生的一些可预测的宏观量来设计个体的策略，并进一步考察个体微观优化行为导致的平均场的形成。数学机器被部署来执行这个项目，包括最优控制理论，偏微分方程，随机过程，等等。方法和结果将感兴趣的应用数学家，经济学家，系统和控制理论家。 该研究计划的另一部分将研究概率环境中的社会舆论动态。这个领域是社会科学，经济学和统计物理学家的极大兴趣。我们的主要兴趣是解决（i）随机不确定性，发生在一个给定的代理人的收购他人的意见和（ii）自由意志引起的噪音，也就是说，一个人的意见可能有随机移位可能是由于人类生理。我们将设计谨慎的意见学习算法在这样的嘈杂的环境中，并研究在个人层面上的简单的意见更新规则所产生的集体模式的形成。这项研究将通过数学建模和分析为理解某些社会和文化现象的模式形成提供新的见解。