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

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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.