This paper presents an inexact proximal method for solving monotone variational inequality problems with a given separable structure. The proposed algorithm is a natural extension of the Proximal Multiplier Algorithm with Proximal Distances (PMAPD) proposed by Sarmiento et al. [Optimization 65 (2016) 501-537], which unified the works of Chen and Teboulle (PCPM method), and Kyono and Fukushima (NPCPMM) developed for solving convex programs with a particular separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle [SIAM J. Optim. 16 (2006) 697-725] with a decomposition method given by Chen and Teboulle for convex problems and extends the results of the Entropic Proximal Decomposition Method proposed by Auslender and Teboulle, which used to Logarithmic Quadratic proximal distances. Under some mild assumptions on the problem we prove a global convergence of the primal-dual sequences produced by the algorithm.
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Acknowledgements. The first and third authors wish to thank at Callao National University and INNOVATE-PERÚ by supporting research through the CONVENIO 460-INNOVATEPERU-BRI-2015-Perú. The work of the second author was supported by the National Council for Scientific and Technological Development (CNPq) and by grant E-26/200.209/2017, Rio de Janeiro Research Foundation (FAPERJ).
© EDP Sciences, ROADEF, SMAI 2021.
- Maximal monotone operators
- Proximal distances
- Separable structure
- Variational inequalities