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Error bounds for proximal point subproblems and associated inexact proximal point algorithms. (English) Zbl 0963.90064

Let \(T\) be a maximal monotone operator on a Hilbert space \(H\) and, given \(\varepsilon\geq 0\), let \(T^\varepsilon\) be the \(\varepsilon\)-enlargement of \(T\) defined by \[ T^\varepsilon(x) =\bigl\{v\in H\mid \forall y\in H,\;u\in T(y),\langle v-u,x-y \rangle\geq- \varepsilon\bigr\}. \] For \(\lambda>O\), the authors consider the proximal system consisting in finding \(y,v\in H\) such that \(v\in T(y)\) and \(\lambda v+y-x=0\), for which they introduce the merit function \[ S_{T,\lambda,x}(y,v)= \|\lambda v+y-x\|^2+2 \lambda \varepsilon_T (y,v), \] with \[ \varepsilon_T(x,v)= \inf\bigl\{ \varepsilon\geq 0 \mid v\in T^\varepsilon (x)\bigr\}. \] This function is strongly convex, a fact that yields an error bound for approximate solutions of the proximal system. Based on this merit function, the authors propose an algorithm to solve the inclusion \(0\leq T(x)\) and study its convergence properties. For the variational inequality problem of finding \(x\in C=H\) such that \(\langle F(x),y-x \rangle\geq 0\) \(\forall y\in C\), with \(F:H\to H\), they obtain an error bound (under suitable assumptions), which involves the mapping \(R_\alpha(x)= x-P_C(x-\alpha F(x))\), \(P_C\) denoting projection onto \(C\) and \(\alpha\) being a sufficiently large positive number.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
47H05 Monotone operators and generalizations
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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