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Relaxed projection and contraction methods for solving Lipschitz continuous monotone variational inequalities

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In each iteration, the projection and contraction (PC) methods need to calculate one (or two) projection. Generally speaking, however, it is difficult to calculate a projection unless the feasible set is simple enough. This might seriously affect the feasibility of the PC methods. To overcome this weakness, in the setting of Hilbert spaces, a relaxed projection and contraction method is proposed for solving Lipschitz continuous and monotone variational inequalities. The core of our algorithm is to replace every projection onto the feasible set with a projection onto some half-space and this makes our algorithm easy to implement. Also, the step length of our algorithm can be selected adaptively. The weak convergence and the O(1 / t) convergence rate of our algorithm are proved. Primary numerical experiments illustrate the performance and advantage of the proposed algorithm.

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Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This work is supported by the Fundamental Research Funds for the Central Universities (3122017078).

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  1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Songnian He, Qiao-Li Dong & Hanlin Tian

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  1. Songnian He
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  2. Qiao-Li Dong
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  3. Hanlin Tian
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Correspondence to Songnian He.

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He, S., Dong, QL. & Tian, H. Relaxed projection and contraction methods for solving Lipschitz continuous monotone variational inequalities. RACSAM 113, 2773–2791 (2019). https://doi.org/10.1007/s13398-019-00658-9

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  • DOI: https://doi.org/10.1007/s13398-019-00658-9

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