Stochastic Optimization: Approximation Algorithms and Asymptotic Analysis

由于新时代技术的快速发展，现实世界的系统已经接受了不断扩大的复杂性。 它们不仅受到随机干扰的影响而连续变化，而且还经历某些切换过程，偶尔会跳变以影响系统的状态，从而呈现另一种不确定性。 针对无线通信、信号处理、制造业、金融和经济等领域的需求，本项目旨在为此类系统设计高效的计算方法和分析特性。 该项目提出的算法考虑到噪音测量和随机环境中的新兴应用在移动的通信和金融市场分析。 受涉及额外环境变量的追捕-逃避游戏的启发，该项目为偶尔和随机切换的游戏开发了数值程序，并可能应用于国土安全。 为了执行识别任务，只有使用传感器获得的数据是可用的，如在汽车工程和医疗应用中，使用量化数据的识别算法将被检查。 为了能够描述复杂系统及其固有的不确定性和随机环境，该项目还强调对随机过程内在特性的理解，包括扩散特征和跳跃特征。 所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。

Owing to the rapid advances in technology in the new era, real-world systems have embraced an ever-expanding complexity. Not only do they vary continuously subject to random disturbances, but also they experience certain switching processes, jump-changing occasionally to affect the systems' states, presenting another fold of uncertainty. In response to the needs in wireless communications, signal processing, manufacturing, finance, and economics, this project aims to design efficient computational methods and analytic properties for such systems. The project presents algorithms taking into consideration noisy measurements and random environments for emerging applications in mobile communications and financial market analysis. Motivated by pursuit-evasion games that involve additional environmental variables, this project develops numerical procedures for games with occasional and random switching, with potential applications to homeland security. To carry out identification tasks where only data obtained using sensors are available, as in automotive engineering and medical applications, identification algorithms using quantized data will be examined. To be able to describe complex systems and their inherent uncertainty and random environment, this project also emphasizes the understanding of the intrinsic properties of random processes, including both diffusive features and jump characteristics. The proposed research will yield new insight, and advance the state of the art of stochastic optimization methods.