Recursive Algorithms and Regime Switching Models for Stochastic Optimization

本研究计画旨在设计随机近似与最佳化演算法，并发展动态系统模式以解决现有与新兴应用所产生的问题。提出了几种具有非光滑动力学或多时间尺度的随机迭代算法，或导致非自治极限常微分方程或微分包含所给出的极限系统。他们的渐近性质，如收敛和收敛速度将通过相关的动力系统进行检查。将制定各种不同的、改进的和有效的程序。所提出的算法跟踪时变参数将导致有限状态切换的普通和随机微分方程，这是不使用现有的方法在随机逼近。 研究离散时间和连续时间系统的马尔可夫链调制的状态切换模型。为了降低复杂性，将使用动态系统的层次结构和时标分离。这些系统的性质将通过聚合和分解方法和奇异摄动方法进行研究。这些性质将进一步被用来指导设计和发展的程序，优化和控制的动态system.To满足日益增长的需求，有效的计算算法和方法的最佳决策，在无线通信，制造系统，金融工程，信号处理，和嵌入式网络，这个项目的目的是设计数学模型有用的现有和新兴的应用，并开发适用于CDMA通信系统、生产计划、均值-方差组合选择和通信网络等问题的算法。为了适应真实的世界中的系统，需要考虑制度的变化。例如，在股票市场的情况下，市场参数取决于在“看涨”和“看跌”状态之间跳跃的市场模式。在这些状态下，相应的市场参数是完全不同的，导致明显不同的行为。此外，股票市场还表现出多时间尺度结构。这样的模型和时标分离也出现在通信网络、生产规划和其他应用中。拟议的研究工作旨在为上述系统设计可行的模型和程序。该研究将产生新的见解，并推进随机优化方法的发展。

This research project is to design stochastic approximation and optimization algorithms, and to develop regime switching dynamic system models for solving problems arising from existing and emerging applications. Several stochastic iterative algorithms featuring non-smooth dynamics or multi-time scales, or leading to non-autonomous limit ordinary differential equations or limit systems given by differential inclusions are proposed. Their asymptotic properties such as convergence and rates of convergence will be examined through the associated dynamic systems. Variants, improvements, and efficient procedures will be developed. The proposed algorithms for tracking time-varying parameters will lead to limit regime-switching ordinary and stochastic differential equations, which are not obtainable using the existing methods in stochastic approximation. Research on regime switching models modulated by Markov chains for both discrete-time and continuous-time systems will be conducted. Aiming at reducing complexity, hierarchical structure of the dynamic systems and time-scale separation will be used. Properties of these systems will be investigated through aggregation and decomposition methods and singular perturbation methodology. These properties will further be used to guide the design and development of procedures for optimization and control of dynamic systems.To meet the growing demand for efficient computational algorithms and methods for optimal decision making in wireless communications, manufacturing systems, financial engineering, signal processing, and queueing networks, this project aims to design mathematical models useful for existing and emerging applications, and to develop algorithms applicable to such problems as CDMA communication systems, production planning, mean-variance portfolio selections, and communication networks. To accommodate systems in the real world, shifts in regimes need to be taken into consideration. Take for instance, the situation in a stock market, the market parameters depend on the market mode that jumps between the "bullish" and "bearish" states. In these states, the corresponding market parameters are quite different resulting in markedly different behavior. In addition, the stock market also exhibits multi-time-scale structure. Such models and time-scale separations also appear in communication networks, production planning and other applications. The proposed research work aims to design feasible models and procedures for the aforementioned systems. The proposed research will yield new insight, and advance state of the art of stochastic optimization methods.