A proximal bundle approach for solving the generalized variational inequalities with inexact data

This paper introduces a proximal bundle scheme to solve generalized variational inequalities with inexact data. Under optimality conditions, the problem can be equivalently represented as seeking out the zero point of the sum of two multi-valued operators whose domains are the real Hilbert space. The two operators denoted by T and f, respectively, are the monotone operator and the subdifferential of a lower semi-continuous, non-differentiable, convex function. Our approach is based on the principles of the proximal point strategy, which involves incorporating inexact information into the subproblems and approximating them using a series of piecewise linear convex functions. Moreover, we put forward a novel stopping criterion to identify the adequacy of the current approximation. This approach serves to make the subproblems more manageable, and it has been proven that obtaining inexact information can ensure that the linearization error during the iteration process remains non-negative, thus avoiding triggering noise attenuation. Subsequently, we verify the convergence of the algorithm under relatively mild assumptions (the operator T is para-monotone and may be multi-valued). Ultimately, we present the findings of elementary numerical experiments to declare the method's efficacy.