Convergence Analysis of the Restarted Moving-Anchored Extra-Gradient Method in the Absence of Local Lipschitz Continuity

In this paper, we introduce the moving-anchored extra-gradient (MAEG) method for solving monotone inclusion problems involving the sum of a continuous monotone operator and a maximal monotone operator. Notably, the distance from the anchor point to the solution set is designed to be monotonically non-increasing. Under Lipschitz continuity of the forward operator, MAEG attains an $\mathcal{O}(1/k)$ non-asymptotic iteration complexity, and when a positive anchor-update parameter is used, it further achieves an $o(1/k)$ asymptotic rate. Furthermore, leveraging the specific behavior of the anchor point, we propose a tailored restart strategy. We demonstrate that this strategy ensures convergence even in the absence of local Lipschitz continuity, while preserving the original iteration complexity guarantees whenever the Lipschitz condition holds.

Source: arXiv:2607.07585v1 - http://arxiv.org/abs/2607.07585v1 PDF: https://arxiv.org/pdf/2607.07585v1 Original Link: http://arxiv.org/abs/2607.07585v1