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Regularization and iterative methods for monotone inverse variational inequalities

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Abstract

We consider the monotone inverse variational inequality: find \(x\in H\) such that

$$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$

where \(\Omega \) is a nonempty closed convex subset of a real Hilbert space \(H\) and \(f:H\rightarrow H\) is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (11226232), the Doctoral Innovation Fund for Young Teacher of the Central Universities (12NZYBS04), the Science Research Fund for the Central Universities (11NPT02) and the Natural Science Foundation of the State Ethnic Affairs Commission of China (SWUN20100706).

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Authors and Affiliations

  1. College of Computer Science and Technology, Southwest University for Nationalities, Chengdu, 610041, Sichuan, People’s Republic of China

    Xue-ping Luo & Jun Yang

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  1. Xue-ping Luo
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Correspondence to Xue-ping Luo.

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Luo, Xp., Yang, J. Regularization and iterative methods for monotone inverse variational inequalities. Optim Lett 8, 1261–1272 (2014). https://doi.org/10.1007/s11590-013-0653-2

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