Abstract
We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.
Highlights
Our focus is on the extension to p-uniformly convex spaces of tools from the analysis of fixed point iterations in linear spaces
We are indebted to the works of Kuwae [26] and Ariza-Ruiz, Leustean, Lopez-Acedo, and Nicolae [2,3] who studied firmly nonexpansive mappings in nonlinear spaces, though the asymptotic behavior of averaged mappings in uniformly convex Banach spaces was already studied by Baillon, Bruck and Reich in [6]
We restrict our attention to an appropriate generalization of single-valued, pointwise α-averaged mappings. This generalization leads to a definition of firmly nonexpansive mappings that is less restrictive than notions with the same name studied in [3,5,15,40,41], though, we show that our notion is implied by the previously studied objects
Summary
Our focus is on the extension to p-uniformly convex spaces of tools from the analysis of fixed point iterations in linear spaces. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to verifying that the individual components of the splitting algorithms satisfy the required regularity. We provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates This allows one to prove convergence of a tremendous variety of splitting algorithms for the first time in spaces with curvature bounded from above. 4 where convergence without rates is established for mappings that are only pointwise nonexpansive at the asymptotic centers of all subsequences (Theorem 27) and quantitative convergence in Theorem 30 under the additional assumption of (gauge) metric subregularity (Definition 29).
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