Let \(H\) be a real Hilbert space and assume that \(C\) is a nonempty closed convex subset of \(H\). Let \(F\) be a monotone multivalued operator defined on \(H\) and consider a lower semicontinuous proper convex function \(\phi :H\rightarrow\mathbb\cup\{+\infty\}\). This paper deals with the study of the following variational inequality: find \(x^*\in C\) and \(r(x^*)\in F(x^*)\) such that, for all \(x\in C\), \(\langle r(x^*),x-x^*\rangle +\phi (x)-\phi (x^*)\geq 0\).
In the first part of the paper the authors introduce a bundle scheme proposed to solve this problem and they prove that if only null-steps are made after some \(x^k\) has been reached, then \(x^k\) actually solves the variational inequality. Next the weak and the strong convergence of the sequence generated by the bundle algorithm is established. Further numerical tests designed to illustrate the behaviour of the bundle algorithm conclude the paper.