Adaptive Projected Subgradient Method for Asymptotic Minimization of Sequence of Nonnegative Convex Functions
Adaptive Projected Subgradient Method for Asymptotic Minimization of Sequence of Nonnegative Convex Functions
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DOI：
10.1081/nfa-200045806
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发表时间：
2005-01
影响因子：
1.2
通讯作者：
I. Yamada;N. Ogura
中科院分区：
文献类型：
--
作者：
I. Yamada;N. Ogura
摘要本文提出了一种自适应投影子梯度法，该算法能在实Hilbert空间上求出闭凸集上的非负凸函数序列的渐近极小值。本文提出的算法是针对固定目标值的非光滑凸优化问题，将Polyak的次梯度算法自然扩展到整个过程中凸目标本身不断变化的情况。主要定理表明了算法的强收敛性和算法生成的序列的渐近最优性，可以作为广泛的非平稳随机过程的集合论自适应滤波方案的统一指导原则。这不仅包括现有的自适应滤波技术；如NLMS、投影NLMS、约束NLMS、APA、自适应并行外投影算法等，以及新技术；例如，自适应并行最小-最大投影算法及其嵌入式约束版本。数值算例表明，该方法适用于鲁棒自适应信号处理问题。
Abstract This paper presents an algorithm, named adaptive projected subgradient method that can minimize asymptotically a certain sequence of nonnegative convex functions over a closed convex set in a real Hilbert space. The proposed algorithm is a natural extension of the Polyak's subgradient algorithm, for nonsmooth convex optimization problem with a fixed target value, to the case where the convex objective itself keeps changing in the whole process. The main theorem, showing the strong convergence of the algorithm as well as the asymptotic optimality of the sequence generated by the algorithm, can serve as a unified guiding principle of a wide range of set theoretic adaptive filtering schemes for nonstationary random processes. These include not only the existing adaptive filtering techniques; e.g., NLMS, Projected NLMS, Constrained NLMS, APA, and Adaptive parallel outer projection algorithm etc., but also new techniques; e.g., Adaptive parallel min-max projection algorithm, and their embedded constraint versions. Numerical examples show that the proposed techniques are well-suited for robust adaptive signal processing problems.