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

摘要

In this paper, we introduce the following m-layered hard constrained convex feasibility problem HCF(m): Find a point u ∈ T_m, where Γ_0 := H (a real Hilbert space), Γ_I := arg min g_I(Γ_(I-1)) and g_I(u) := ∑ from j=1 to M_I of w_(I,j d)~2 (u,C_(I,j)) are defined for (ⅰ) nonempty closed convex sets C_(I,j) is contained in H and (ⅱ) weights w_(I,j) > 0 satisfying ∑ from j=1 to M_I of w_(I,j) = 1 (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 ∈ ∩ from I=1 to M of C_I ≠ φ, where C_I is contained in 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) = φ, 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_(I,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_(1,1) or C_(2,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.