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Convergence rate analysis of an operator splitting method for solving a class of variational inequality problems. (Chinese. English summary) Zbl 1463.65162

Summary: This paper considers a class of variational inequality problems: finding \(\boldsymbol{x}^* \in \Omega\), such that \(\boldsymbol{F} (\boldsymbol{x}^*)^{\mathrm{T}} (\boldsymbol{x} - \boldsymbol{x}^*) \ge 0\), \(\forall \boldsymbol{x} \in \Omega\), where \(\Omega \subseteq \mathbb{R}^n\) is nonempty, closed and convex, \(\boldsymbol{F} = \boldsymbol{f} + \boldsymbol{g}\) is a continuous mapping from \(\mathbb{R}^n\) to \(\mathbb{R}^n\), \(\boldsymbol{f}\) and \(\boldsymbol{g}\) are monotone but \(\boldsymbol{f}\) is unknown. We study an operator splitting method for this class of problems with a variety of applications. Based on the previous convergence results, we further analyze the \(O (1/k)\) and \(o (1/k)\) sublinear convergence rate in non-ergodic sense for this operator splitting method, where \(k\) counts the iteration number. Finally, numerical results demonstrate the efficiency of the algorithm.

MSC:

65K15 Numerical methods for variational inequalities and related problems
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