Abstract
Inspired by the very recent results of Wang and Xu (2010), we study properties of the approximating curve with 1-norm regularization method for the split feasibility problem (SFP). The concept of the minimum-norm solution set of SFP in the sense of 1-norm is proposed, and the relationship between the approximating curve and the minimum-norm solution set is obtained.
Highlights
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively
It is well known that SFP 1.1 is equivalent to the minimization problem min x∈C
H1 and H2 in SFP 1.1 are restricted to RN and RM, respectively, and · will stand for the usual 2-norm of any Euclidean space Rl; that is, for any x x1, x2, . . . , xl ∈ Rl, x x12 · · · xl[2]
Summary
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively. Let PC denote the projection from H onto a nonempty closed convex subset C of H; that is, PCx arg min x − y , x ∈ H. The following lemma gives the optimality condition for the minimizer of a convex functional over a closed convex subset.
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