Convex Feasibility Problem with Prioritized Hard Constraints : Double Layered Projected Gradient Method(Numerical Analysis and Optimization)

概要

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In this paper, we introduce the following m-layered hard constrained convex feasibility problem HCF(m): Find a point u ∈ Γ_m, where Γ_O := H(a real Hilbert space), Γ_i := argmin g_i(Γ_<i-1>) and g_i(u) := Σ^<M_i>_<j=1> W_<i,j>d^2(u, C_<i,j>) are defined for (i) nonempty closed convex sets C_<i,j> ⊂ H and (ii) weights w_<i,j> > 0 satisfying Σ^<M_i>_<j=1> w_<i,j> = I (i ∈ {1,・・・,m}, j ∈ {1,・・・,M_i}). This problem is regarded as a natural extension of the standard convex feasibility problem: find a point u ∈ ∩^M_<i=1> C_i ≠ 0, where C_i ⊂ H (i ∈ {1,・・・,M}) are closed convex sets. Unlike the standard problem, HCF(m) can handle the inconsistent case; i.e., ∩_<i,j>C_<i,j> = 0 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 C,j, which generates a sequence (u_n) satisfying lim_<n→∞> d(u_n,Γ_3) = 0 (for M_1 = 1) when at least one of C_<i,j> or C_<2,j> 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.
社団法人電子情報通信学会の論文
2004-04-01