Optimization over the fixed point set of nonexpansive mapping and its application

非扩张映射不动点集的优化及其应用

• 批准号: 10650350
• 负责人: YAMADA Isao
• 金额: $ 0.51万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10650350/
• 关键词: Hybrid steepest descent method; fixed point; nonexpansive mapping; convex projection; POCS; blind image deconvolution; inverse problem; neural network

The convex projection algorithm is a class of algorithms finding a point, with convex projections, in the intersection of multiple closed convex sets. The basic idea of the algorithms originated from J. von Neumann's alternating projection in 1933. Although the point obtained by the method is only guaranteed to belong to the intersection of given closed convex sets, the remarkable effect and the universal applicability of the simple algorithms have been commonly recognized in many branches of applied mathematical, physical, computer sciences and engineerings since the algorithm POCS was successfully applied to the image restoration problem by D.C. Youla and H. Webb in 1982.In this research project, we develop a new algorithm mamed Hybrid steepest descent method that minimizes a given convex cost function over the fixed point set of a nonexpansive mapping in a real Hilbert space. The nonexpansive mapping is extremly general class of mappings including the convex projection. By this great generality, many open problems, in signal processing, not handled by the standard convex projection technique have become resolved. Indeed we successfully applied the method to the following important problems:1. Approximation of convexly constrained pseudoinverse operator,2. Constrained least squares design of M-D FIR filter,3. Design of two channel linear phase FIR QMF banks,4. Set-theoretic blind image deconvolution problem,5. Design of associative memory neural network to recall nearest pattern from input.

凸投影算法是一类算法，在多个闭合凸集的相交中找到了一个带有凸投影的点。该算法的基本思想起源于1933年的J. von Neumann的交替预测。尽管该方法获得的点仅保证属于给定的封闭凸集集合的相交，但简单算法的显着效果和普遍的适用性是在应用程序，物理上的许多分支中都认识到了计算机和工程师的许多分支。 D.C. Youla和H. Webb的图像恢复问题于1982年。在该研究项目中，我们开发了一种新的算法杂交混合陡峭的下降方法，该方法将给定的凸成本函数最小化，这是在真实希尔伯特空间中无用映射的固定点集中的给定凸成本函数。非专用映射是包括凸投影在内的极端映射类别。通过这种伟大的一般性，在信号处理中，许多未经标准凸投影技术处理的开放问题已经解决。确实，我们成功地将方法应用于以下重要问题：1。凸限制的伪算子的近似值，2。限制了M-D FIR滤波器的最小二乘设计，3。两个通道线性相位FIR QMF银行的设计，4。设定理论盲图片反卷积问题，5。联想记忆神经网络的设计，以回忆输入最接近模式。

项目成果

F.Deutsch and I.YAMADA: "Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings" Numerical Functional Analysis and Optimization. 19 1&2. 33-56 (1998)

F.Deutsch 和 I.YAMADA：“最小化非扩张映射的定点集交集上的某些凸函数”数值泛函分析和优化。

I. Yamada, S. Iine, K. Sakaniwa: "An Associative Memory Neutral Network to Recall Nearest Pattern from Input"IEICE Transactions Fundamentals. E82-A. 2811-2817 (1999)

I. Yamada、S. Iine、K. Sakaniwa：“从输入中调用最近模式的联想记忆中性网络”IEICE 交易基础知识。

Masanori KATO, Isao YAMADA, and Kohichi SAKANIWA: "An Optimal Set-theoretic Blind Deconvolution Scheme based on Hybrid Steepest Descent Method"Proceedings of the 1998 International Symposium on Information Theory and Its Applications. 405-409 (1999)

Masanori KATO、Isao YAMADA 和 Kohichi SAKANIWA：“基于混合最速下降法的最优集合论盲解卷积方案”1998 年信息论及其应用国际研讨会论文集。

Masanori, KATO, Isao YAMADA, and Kohichi SAKANIWA: "An optical blind deconvolution scheme based on convex projection techniques"Proceedings of the 1998 International Symposium on Information, Theory and Its Applications. 219-222 (1998)

Masanori、KATO、Isao YAMADA 和 Kohichi SAKANIWA：“基于凸投影技术的光学盲反卷积方案”1998 年国际信息、理论及其应用研讨会论文集。

Isao YAMADA: "Approximation of Convexly Constrained Pseudoinverse by Hybrid Steepest Descent Method (Invited)"Proceedings of the 1999 International Symposium on Circuits and Systems Vol. V, 37-40, Orlando FL, June. (1999)

Isao YAMADA：“通过混合最速下降法近似凸约束伪逆（特邀）”1999 年国际电路与系统研讨会论文集第 1 卷。