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Weak and strong convergence of generalized proximal point algorithms with relaxed parameters

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Abstract

In this work, we propose and study a framework of generalized proximal point algorithms associated with a maximally monotone operator. We indicate sufficient conditions on the regularization and relaxation parameters of generalized proximal point algorithms for the equivalence of the boundedness of the sequence of iterations generated by this algorithm and the non-emptiness of the zero set of the maximally monotone operator, and for the weak and strong convergence of the algorithm. Our results cover or improve many results on generalized proximal point algorithms in our references. Improvements of our results are illustrated by comparing our results with related known ones.

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Notes

  1. As is the custom, in the whole work, we use the empty sum convention and empty product convention, that is, given a sequence \((t_{k})_{ k\in {\mathbb {N}}}\) in \({\mathbb {R}}\), for every m and n in \({\mathbb {N}}\) with \(m > n\), we have \(\sum ^{n}_{i=m} t_{i} =0\) and \(\prod ^{n}_{i=m} t_{i} =1\).

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  1. Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Hui Ouyang

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  1. Hui Ouyang
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Ouyang, H. Weak and strong convergence of generalized proximal point algorithms with relaxed parameters. J Glob Optim 85, 969–1002 (2023). https://doi.org/10.1007/s10898-022-01241-0

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  • DOI: https://doi.org/10.1007/s10898-022-01241-0

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