"Nonsmooth dynamics associated to variational inequalities, generalized Nash games and applications"

“与变分不等式相关的非光滑动力学、广义纳什博弈和应用”

• 批准号: 262899-2012
• 负责人: Cojocaru, MonicaGabriela
• 金额: $ 1.09万
• 依托单位: University of Guelph
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571300
• 关键词: Nonsmooth; dynamics; associated; variational; inequalities

This proposal investigates the relationship between three important mathematical constructs: nonsmooth dynamical systems, quasivariational inequalities, and generalized Nash games, and their appropriate applications. Variational inequalities (VI) were introduced in the early 60s within the framework of partial differential equations. Later it was shown they equivalently reformulate large classes of equilibrium problems (such as Nash, Wardrop, Walras, Cournot, traffic equilibrium etc.), thus becoming prominent in operations research. Due to this equivalence, solutions of VI are often called equilibria. Of specific interest in my work is a particular class of VI, called "quasivariational" (QVI). Essentially a QVI models equilibrium problems with constraint sets dependent on the equilibrium solution. At their core VI/QVI problems are static. To describe the dynamics of such problems, approaches range from inequalities of evolution to recent concepts of evolutionary and differential variational inequalities. My approach relies on associating VI/QVI to dynamical systems given by solutions to a differential equation or inclusion so that two conditions are met: the solutions of the VI/QVI coincide with the equation's or inclusion's stationary points; the solutions of the equation/inclusion remain inside the constraint set at all times. These systems belong to the area of nonsmooth dynamics. Results from QVI theory imply existence of stationary solutions for the associated dynamics. However, existence, uniqueness and stability analysis of non-stationary solutions is largely an open problem. Investigating these questions is my first objective. This objective is a natural extension of some of my previous work relating VI with nonsmooth systems. My second objective is to use results above to provide novel ways to solve generalized Nash (GN) games which can be equivalently reformulated as QVI problems. Solutions for some GN games are known, but in general this is an open question. My third objective concerns applications of results above in dynamic population models of decision making, in time-dependent traffic network problems with changes in network topology, and in epidemiology.

该提案研究了三个重要数学结构之间的关系：非光滑动力系统、拟变分不等式和广义纳什博弈及其适当的应用。变分不等式 (VI) 于 20 世纪 60 年代初在偏微分方程的框架内引入。后来的研究表明，他们等效地重新表述了大类均衡问题（例如纳什、沃德罗普、瓦尔拉斯、古诺、交通均衡等），从而在运筹学中变得突出。由于这种等价性，VI 的解通常称为均衡。我的工作中特别感兴趣的是一类特殊的 VI，称为“准变分”(QVI)。本质上，QVI 对平衡问题进行建模，其约束集取决于平衡解。 VI/QVI 问题的核心是静态的。为了描述此类问题的动态，方法范围从进化不等式到最新的进化和微分变分不等式概念。我的方法依赖于将 VI/QVI 与由微分方程或包含的解给出的动态系统相关联，从而满足两个条件：VI/QVI 的解与方程或包含的驻点一致；方程/包含式的解始终保持在约束集内。这些系统属于非光滑动力学领域。 QVI 理论的结果意味着相关动力学存在平稳解。然而，非平稳解的存在性、唯一性和稳定性分析很大程度上是一个悬而未决的问题。调查这些问题是我的首要目标。这个目标是我之前将 VI 与非光滑系统相关的一些工作的自然延伸。我的第二个目标是利用上述结果提供解决广义纳什（GN）博弈的新颖方法，这些博弈可以等效地重新表述为 QVI 问题。一些 GN 游戏的解决方案是已知的，但总的来说这是一个悬而未决的问题。我的第三个目标涉及上述结果在决策的动态群体模型、网络拓扑变化的时间相关交通网络问题以及流行病学中的应用。