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On the Nonexpansive Operators Based on Arbitrary Metric: A Degenerate Analysis

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Abstract

We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is discussed with particular focus on the case of degenerate metric, since the degeneracy is often encountered when reformulating many existing first-order operator splitting algorithms as a metric resolvent. This work paves a way for analyzing the generalized proximal point algorithm with a non-trivial relaxation step and degenerate metric.

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Notes

  1. The ‘non-trivial’ means that the relaxation operator is not multiple of identity operator, see [26, Eq.(2.8)], [25, Eq.(3.5)] and [24, Eq.(5.2)] for example.

  2. Refer to [5, Lemma 2.47] or [1, Lemma 2.1] for the Opial’s argument.

  3. This line of reasoning is very similar to Fejér monotonicity, see [5, Proposition 5.4, Theorem 5.5] for example.

  4. It is also called F-resolvent in [4] or warped resolvent [13].

  5. Here, the functions f and g are assumed to be proper, lower semi-continuous and convex.

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  1. National Key Laboratory, Beijing, China

    Feng Xue

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Xue, F. On the Nonexpansive Operators Based on Arbitrary Metric: A Degenerate Analysis. Results Math 77, 237 (2022). https://doi.org/10.1007/s00025-022-01766-6

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