An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions (Q2757632)

The article under review is a valuable contribution to the algorithmical treatment of variational inequality problems VIP\((T,C)\): NEWLINE\[NEWLINEx^*\in C,\;v^*\in T(x^*),\;\langle x-x^*,v^*\rangle\geq 0\quad \forall x\in C,NEWLINE\]NEWLINE where \(C\) is a closed convex set and \(T\) is a maximal monotone set-valued operator in \(\mathbb{R}^n\). In research done by numerous authors before, the ``classical'' iterative method consisted in sequences \((x^k)\) where the \(x^k\) are solutions of proximal subproblems defined using \(x^k\). Now, the new generalized proximal point method consists in stepwise solution of subproblems where gradients of so-called Bregman functions \(f\) (by definition, strictly convex, differentiable in \(\text{int} (C)\) and with gradient diverging at \(\partial C)\) are incorporated. Slightly perturbing these subproblems, the solutions are called inexact. Two basic assumptions consist in the existence of an interior (in \(C)\) solution of the generalized proximal subproblem, and in boundary coerciveness of \(f\). The precise algorithmical steps of selection and choice of parameters are performed using Bregman distance (on \(f)\). This is analytically well-prepared and new, and it serves for prescribing an error tolerance.NEWLINENEWLINENEWLINEProvided that VIP\((T,C)\) has a solution, and that this is of interior position or that \(T\) is paramonotone (e.g., maximal monotone), the main result says that \((x^k)\) converges to a solution of VIP\((T,C)\). In the paramonotone case, the authors have got rid of ``paramonotonicity'' assumption made in literature before. Moreover, as a consequence of other properties of Bregman functions, the classical assumption of ``convergence consistency'' has also become superfluous. The article contains a careful convergence analysis, and it is well-readable.