The new inexact bundle-type technique to solve variational inequalities of composite structures

In this research, we introduce an innovative composite proximal bundle method aimed at resolving generalized variational inequalities with the integration of inexact oracles. The essence of the problem is distilled into the pursuit of the null space of the superposition of two multivalued operators, all within the realm of a real Hilbert space and underpinned by an optimality criterion. We define the non-differentiable composite convex function and the monotone operator as f and Γ, respectively, both characterized by their roles as subdifferentials of functions that are lower semicontinuous. Central to our methodology is the proximal point approach, which entails the iterative refinement of subproblems through the lens of piecewise linear convex functions, augmented by the incorporation of inexact data. To enhance the computational efficiency and precision, we introduce a novel stopping criterion, designed to evaluate the precision of the current approximation. This innovation is pivotal in streamlining the management of subproblems. Moreover, to safeguard the convergence properties of our algorithm against the potential perturbations induced by inexact data, we have integrated a denoising technique. Subsequent to these enhancements, we demonstrate the convergence of our algorithm under specific operator properties, thereby providing a robust theoretical underpinning for its application. To verify the practical efficacy of our approach, we present the outcomes of our numerical experiments, which affirm the effectiveness of our method in the context of non-smooth optimization.