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The Landweber Operator Approach to the Split Equality Problem
【Abstract】 The split equality problem (SEP) seeks a pair of points $(x^{\ast },y^{\ast })\in (C,D)$ with the property that $Ax^{\ast }=By^{\ast }$, where $C,D$ are nonempty closed convex subsets of Hilbert spaces $\mathcal{H}_{1}$ and $% \mathcal{H}_{2}$, respectively, and $A:\mathcal{H}_{1}\rightarrow \mathcal{H}% _{3}$ and $B:\mathcal{H}_{2}\rightarrow \mathcal{H}_{3}$ are bounded linear operators, where $\mathcal{H}_{3}$ is another Hilbert space. The SEP can equivalently be converted to a split feasibility problem in the product space $\mathcal{H}_{1}\times \mathcal{H}_{2}$. Using this equivalence, we are able to provide a Landweber operator approach to studying the convergence of several iterative methods for finding a solution to the SEP. We also discuss the linear regularity of the Landweber operator associated with the SEP and linear convergence of the iterative methods.
【Author】 Hong-Kun Xu, Andrzej Cegielski
【Keywords】 Landweber operator,nonexpansive mapping,averaged mapping,projection,split feasibility,split equality,linear regularity,linear convergence,47J25,47H09,47H10,49M05,90C25
【Journal】 SIAM Journal on Optimization(IF:2.9) Time:2021-02-23
【DOI】 10.1137/20M1337910 [Quote]
【Link】 Article PDF
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