Mathematical Sciences: Problems in Stochastic Optimization

该项目将从提高收敛速度的递归算法、分布参数系统识别过程的渐近性质、自适应滤波算法和宽带噪声微分延迟系统的近最优控制的角度研究随机优化。在相关噪声处理下，分析了一种有吸引力且易于实现的随机逼近算法。其思想是在不使用“预白化”的情况下，对经典近似方法中获得的迭代求平均值。在处理分布式系统的识别算法时，将采用函数不变性方法。所得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，将研究带有延迟的算法和使用“符号”的投影过程。给出了收敛概率为1的充分条件。对于受随机扰动的小参数微分时滞方程，将采用弱收敛方法和松弛控制的方法来发展近似最优控制。这些研究的应用范围从金融市场的分析到更精确的雷达软件的构建。

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.