A modified inertial self-adaptive algorithm for bilevel variational inequalities with applications to signal processing

In this paper, we develop a modified inertial triple self-adaptive algorithm that integrates the inertial method, the subgradient extragradient method, and the hybrid steepest descent method for solving bilevel variational inequality problems, where the upper-level problem involves a strongly monotone variational inequality problem and the lower-level problem comprises a quasimonotone variational inequality problem with split variational inequalities and fixed-point constraints. A distinctive feature of our approach is a triple self-adaptive step-size mechanism that dynamically adjusts parameters without requiring prior estimation of operator norms or Lipschitz constants. We rigorously establish the strong convergence theory for the algorithm under the bilevel framework. Experimental results on two numerical examples and signal recovery applications, including the modified LASSO problem, demonstrate that the proposed algorithm achieves superior performance compared to existing methods in terms of both computational efficiency and practical applicability.