Recursive Algorithms and Regime Switching Models for Stochastic Optimization

本研究项目旨在设计随机逼近和优化算法，并开发状态切换动态系统模型，以解决现有和新兴应用中出现的问题。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，或导致由微分包含给出的非自治极限常微分方程或极限系统。它们的渐近性质，如收敛性和收敛率将通过相关的动态系统来检验。将开发各种变体、改进和有效的程序。所提出的时变参数跟踪算法将导致极限状态切换的常微分方程和随机微分方程，这是用现有的随机逼近方法无法得到的。对离散和连续系统的马尔可夫链调制的状态切换模型进行了研究。为了降低复杂性，将采用动态系统的分层结构和时间尺度分离。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。这些特性将进一步用于指导动态系统优化和控制程序的设计和开发。为了满足无线通信、制造系统、金融工程、信号处理和排队网络对高效计算算法和最优决策方法日益增长的需求，本项目旨在设计对现有和新兴应用有用的数学模型，并开发适用于CDMA通信系统、生产计划、均值方差投资组合选择和通信网络等问题的算法。为了适应现实世界中的系统，需要考虑到制度的变化。以股票市场为例，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，相应的市场参数差异很大，导致了明显不同的行为。此外，股票市场也表现出多时间尺度结构。这种模型和时间尺度分离也出现在通信网络、生产计划和其他应用中。本文的研究工作旨在为上述系统设计可行的模型和程序。提出的研究将产生新的见解，并推进随机优化方法的艺术状态。

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.