Mathematical Sciences: Problems in Stochastic Optimization

本项目将从收敛速度提高的递归算法、分布参数系统辨识过程的渐近性质、自适应滤波算法和具有宽带噪声的差分时滞系统的近最优控制的角度来研究随机优化。在相关噪声过程下，分析了一种具有吸引力且易于实现的随机逼近算法。其思想是取在经典近似方法中获得的迭代的平均值，而不使用“预白化”。对于分布式系统的辨识算法，将采用函数不变性的方法。由此得到的性质将有助于更好地理解收敛速度问题。在自适应滤波问题中，将研究具有延迟的算法和使用“符号”的投影过程。给出了概率为1收敛的充分条件。利用弱收敛方法和松弛控制方法，给出了具有随机扰动的小参数时滞微分方程的近似最优控制。这些研究的应用范围从金融市场分析到为雷达建造更精确的软件。

This project will study stochastic optimization from the point of view of recursive algorithms with improved convergence speed, asymptotic properties of identification procedures for distributed parameter systems, algorithms for adaptive filtering and nearly optimal controls for differential delay systems with wideband noise. An attractive and easily implementable stochastic approximation algorithm will be analyzed under correlated noise processes. The idea is to take the average of the iterates obtained in the classical approximation approach without using "prewhitening". Dealing with the identification algorithms for distributed systems, a functional invariance approach will be taken. The resulting properties will help to get a better understanding on the rate of convergence issue. In the adaptive filtering problem, algorithms with delays and projection procedures using "sign" will be investigated. Sufficient conditions for with probability one convergence will be derived. Nearly optimal controls for differential delay equations subject to random perturbations and with a small parameter will be developed by means of weak convergence methods and the use of relaxed controls. Applications of these studies are found in areas ranging from the analysis of financial markets to the construction of more precise software for radars.