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

凸投影算法是在多个闭凸集的交点上寻找具有凸投影的点的一类算法。该算法的基本思想起源于1933年冯·诺伊曼的交替投影理论。虽然该方法得到的点只能保证属于给定闭凸集的交点，但自1982年D.C. Youla和H. Webb成功地将POCS算法应用于图像恢复问题以来，简单算法的显著效果和普遍适用性在应用数学、物理、计算机科学和工程的许多分支中得到了普遍认可。在本研究项目中，我们开发了一种新的算法，称为混合最陡下降法，该算法在实Hilbert空间的非扩张映射的不动点集上最小化给定的凸代价函数。非膨胀映射是包含凸投影的极为一般的一类映射。由于这种通用性，在信号处理中，许多标准凸投影技术无法解决的开放性问题都得到了解决。事实上，我们成功地将该方法应用于以下重要问题：凸约束伪逆算子的近似，2。2 . M-D FIR滤波器的约束最小二乘设计；2通道线性相位FIR QMF组的设计；集合论盲图像反卷积问题，5。联想记忆神经网络的设计与记忆。

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.