Burachik, Regina S.; Iusem, Alfredo N.; Svaiter, B. F. Enlargement of monotone operators with applications to variational inequalities. (English) Zbl 0882.90105 Set-Valued Anal. 5, No. 2, 159-180 (1997). Summary: Given a point-to-set operator \(T\), we introduce the operator \(T^\varepsilon\) defined as \(T^\varepsilon(x)= \{u: \langle u-v,x- y\rangle\geq -\varepsilon\) for all \(y\in\mathbb{R}^n\), \(v\in T(y)\}\). When \(T\) is maximal monotone \(T^\varepsilon\) inherits most properties of the \(\varepsilon\)-subdifferential, e.g. it is bounded on bounded sets, \(T^\varepsilon(x)\) contains the image through \(T\) of a sufficiently small ball around \(x\), etc. We prove these and other relevant properties of \(T^\varepsilon\), and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form \(0\in H^\varepsilon(x)\), while the subproblems of the exact method are of the form \(0\in H(X)\). If \(\varepsilon_k\) is the coefficient used in the \(k\)th iteration and the \(\varepsilon_k\)’s are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved. Cited in 1 ReviewCited in 92 Documents MSC: 90C25 Convex programming 90C30 Nonlinear programming Keywords:convex optimization; variational inequalities; proximal point methods; monotone operators; point-to-set operator; \(\varepsilon\)-subdifferential PDFBibTeX XMLCite \textit{R. S. Burachik} et al., Set-Valued Anal. 5, No. 2, 159--180 (1997; Zbl 0882.90105) Full Text: DOI