Non-Lipschitz Inertial Contraction-Type Method for Monotone Variational Inclusion problems

TL;DR

This paper introduces a novel inertial contraction-type method for monotone variational inclusion problems that does not require Lipschitz continuity or coercivity, achieving weak and strong convergence with numerical validation.

Contribution

The proposed method relaxes common assumptions, allowing convergence without Lipschitz or coercivity conditions, and demonstrates both weak and strong convergence with linear rate.

Findings

Achieved weak convergence with rate O(1/√k).

Established strong convergence with linear rate under monotonicity.

Validated effectiveness through numerical experiments on signal recovery.

Abstract

This study explores an inertial-based contraction-type approach for addressing monotone variational inclusion problems (in short, MVIP) within real Hilbert spaces. Most contraction-type techniques assume Lipschitz continuity and monotonicity or co-coercivity (inverse strongly monotone) of the single-valued operator. However, the key advantage of the proposed method is that it does not rely on the coercivity condition and the Lipschitz continuity for the single-valued operator. A weak convergence result has been achieved for the proposed algorithm with a convergence rate $\mathcal{O}\left(1/\sqrt{k}\right)$. In addition, the maximal and strong monotonicity of the set-valued operator is used to establish a strong convergence result with the linear convergence rate. To demonstrate the effectiveness of our proposed method, we conduct numerical experiments focused on signal recovery problems.