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Enlargement of monotone operators with applications to variational inequalities. (English) Zbl 0882.90105

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.

MSC:

90C25 Convex programming
90C30 Nonlinear programming
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