Projection and contraction methods with double inertial steps for variational inclusion problems on Hilbert spaces

Moin Uddin; Mohammed Alshahrani; Qamrul Hasan Ansari

doi:10.1016/j.cam.2026.117993

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arXiv · 2607.25203

Projection and contraction methods with double inertial steps for variational inclusion problems on Hilbert spaces

Moin Uddin·Mohammed Alshahrani·Qamrul Hasan Ansari

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Abstract

In this paper, we propose three projection--contraction algorithms for solving variational inclusion problems in the setting of Hilbert spaces, each incorporating a double inertial technique into the projection--contraction framework. The first algorithm \algoO achieves weak convergence under monotonicity and Lipschitz continuity of the single-valued operator, with an adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. A modified variant \algoT of \algoO attains $R$-linear convergence under the strong monotonicity assumption. The third algorithm \algoDIM obtains strong convergence to the minimum-norm solution without requiring strong monotonicity. We illustrate the proposed methods on an abstract variational inclusion problem and apply them to the split feasibility problem and the elastic net regularization problem, comparing them with existing methods in the literature. The $R$-linear convergence of \algoT is also verified through numerical simulations.

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Moin Uddin, Mohammed Alshahrani, Qamrul Hasan Ansari. 2026-07-28. Projection and contraction methods with double inertial steps for variational inclusion problems on Hilbert spaces. https://doi.org/10.1016/j.cam.2026.117993

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