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Extrapolated adaptive proximal method with a single dynamic parameter for variational inequalities and optimization applications
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DOI：10.1016/j.cnsns.2026.110495.png)
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Abstract
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The paper presents an adaptive extrapolated proximal algorithm for variational inequality problems, which also encompasses composite minimization settings. At each iteration, the algorithm requires only a single evaluation of the underlying operator and the corresponding proximal mapping, while utilizing information from the two previous iterates. The method relies on a single dynamic parameter that varies with the iterations and controls the behavior of the generated sequences. Unlike many existing extrapolation schemes, this parameter is independent of the iterates and allows a flexible range of choices. We provide sufficient conditions to guarantee convergence of the generated sequences, even when the underlying operator is not necessarily Lipschitz continuous, and establish that the convergence rate is at least linear. The case where the operator is Lipschitz continuous is recovered as a special case. The effectiveness of the proposed method is illustrated through applications to optimal control, sparse logistic regression, and data classification via Lasso-regularized Extreme Learning Machines. The numerical results demonstrate the robustness and efficiency of the algorithm in solving composite optimization problems. The obtained results pave the way for a new direction of extrapolation in which iterative algorithms for optimization can be developed.
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