A Two-Step Inertial Method for Common Variational Inclusions with Applications to Sparse Learning Models

This paper investigates common variational inclusion problems (CVIPs) beyond the traditional inverse strongly monotone setting, focusing instead on a broader class of monotone and Lipschitz continuous operators. We introduce a novel two-step inertial algorithm incorporating self-adaptive regularization and relaxation techniques to address this generalized framework. We establish the strong convergence of the proposed method under mild assumptions. Extensive numerical experiments demonstrate that our algorithm consistently outperforms several existing approaches. Furthermore, we highlight the practical significance of our framework by reformulating a range of real-world applications, such as the elastic net in statistical learning, sparse logistic regression, and LASSO, as instances of CVIPs.