On the proximal point algorithm. (English) Zbl 1144.49007
Summary: Let \(A\) be a maximal monotone operator in a real Hilbert space \(H\) and let \(\{u_n\}\) be the sequence in \(H\) given by the proximal point algorithm, defined by \(u_n =(I+c_n A)^{ - 1}(u_{n - 1} - f_n ), \forall n\geq 1\), with \(u _{0}=z\), where \(c_n >0\) and \(f_n \in H\). We show, among other things, that under suitable conditions, \(u_n\) converges weakly or strongly to a zero of \(A\) if and only if \(\lim \inf_{n\rightarrow +\infty} |w_n |<+\infty\), where \(w_n =(\sum _{k=1}^n c_k )^{ - 1}\sum _{k=1}^n c_k u_k\) . Our results extend previous results by several authors who obtained similar results by assuming \(A ^{ - 1}(0)\neq \emptyset \).
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H05 | Monotone operators and generalizations |
Keywords:
proximal-point algorithms; variational inequalities; ergodic theorems; maximal monotone operators; asymptotic centersReferences:
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