## 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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