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Extragradient-type methods with $$\mathcal {O}\left( 1/k\right) $$ O 1 / k last-iterate convergence rates for co-hypomonotone inclusions

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  • Quoc Tran-Dinh

    (The University of North Carolina at Chapel Hill)

Abstract

We develop two “Nesterov’s accelerated” variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng’s forward-backward-forward splitting (FBFS) method, while the second one is a Nesterov’s accelerated variant of the “past” FBFS scheme, which requires only one evaluation of the Lipschitz operator and one resolvent of the multivalued mapping. Under appropriate conditions on the parameters, we theoretically prove that both algorithms achieve $$\mathcal {O}\left( 1/k\right) $$ O 1 / k last-iterate convergence rates on the residual norm, where k is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type methods for root-finding problems. For comparison, we also provide a new convergence analysis of the two recent extra-anchored gradient-type methods for solving co-hypomonotone inclusions.

Suggested Citation

  • Quoc Tran-Dinh, 2024. "Extragradient-type methods with $$\mathcal {O}\left( 1/k\right) $$ O 1 / k last-iterate convergence rates for co-hypomonotone inclusions," Journal of Global Optimization, Springer, vol. 89(1), pages 197-221, May.
    • Handle: RePEc:spr:jglopt:v:89:y:2024:i:1:d:10.1007_s10898-023-01347-z
    DOI: 10.1007/s10898-023-01347-z
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    References listed on IDEAS

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    1. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, March.
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    3. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
    4. Cong Dang & Guanghui Lan, 2015. "On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators," Computational Optimization and Applications, Springer, vol. 60(2), pages 277-310, March.
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