Convex Feasibility Problem with Prioritized Hard Constraints--Double Layered Projected Gradient Method

Nobuhiko OGURA  Isao YAMADA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E87-A   No.4   pp.872-878
Publication Date: 2004/04/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Numerical Analysis and Optimization
convex projection,  convex feasibility problem,  hard constraint,  nonexpansive mapping,  fixed point theorem,  

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In this paper, we introduce the following m-layered hard constrained convex feasibility problem HCF(m): Find a point u m, where 0:=H (a real Hilbert space), i: = arg min gi(i-1) and gi(u):=wi,jd 2(u,Ci,j) are defined for (i) nonempty closed convex sets Ci,jH and (ii) weights wi,j > 0 satisfying wi,j=1 (i {1,,m}, j {1,,Mi}. This problem is regarded as a natural extension of the standard convex feasibility problem: find a point u Ci, where Ci H (i {1,, M}) are closed convex sets. Unlike the standard problem, HCF(m) can handle the inconsistent case; i.e., i,j Ci,j = , which unfortunately arises in many signal processing, estimation and design problems. As an application of the hybrid steepest descent method for the asymptotically shrinking nonexpansive mapping, we present an algorithm, based on the use of the metric projections onto Ci,j, which generates a sequence (un) satisfying limn d(un,3) = 0 (for M1 = 1) when at least one of C1,1 or C2,j's is bounded and H is finite dimensional. An application of the proposed algorithm to the pulse shaping problem is given to demonstrate the great flexibility of the method.