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A Direct Splitting Method for Nonsmooth Variational Inequalities

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  • J. Y. Bello Cruz

    (IME, Federal University of Goiás, Campus Samambaia, CEP 74001-970)

  • R. Díaz Millán

    (Federal Institute of Education, Science and Technology)

Abstract

We propose a direct splitting method for solving a nonsmooth variational inequality in Hilbert spaces. The weak convergence is established when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which strongly explores the structure of the operator using projected subgradient-like techniques. The advantage of our method is that any nontrivial subproblem must be solved, like the evaluation of the resolvent operator. The necessity to compute proximal iterations is the main difficulty of other schemes for solving this kind of problem.

Suggested Citation

  • J. Y. Bello Cruz & R. Díaz Millán, 2014. "A Direct Splitting Method for Nonsmooth Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 728-737, June.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:3:d:10.1007_s10957-013-0478-2
    DOI: 10.1007/s10957-013-0478-2
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    References listed on IDEAS

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    1. Alfredo N. Iusem & B. F. Svaiter & Marc Teboulle, 1994. "Entropy-Like Proximal Methods in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 790-814, November.
    2. J. Bello Cruz & A. Iusem, 2010. "Convergence of direct methods for paramonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 46(2), pages 247-263, June.
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    Cited by:

    1. J. Y. Bello Cruz & R. Díaz Millán, 2016. "A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces," Journal of Global Optimization, Springer, vol. 65(3), pages 597-614, July.
    2. Trinh Ngoc Hai, 2020. "Two modified extragradient algorithms for solving variational inequalities," Journal of Global Optimization, Springer, vol. 78(1), pages 91-106, September.
    3. R. Díaz Millán & M. Pentón Machado, 2019. "Inexact proximal $$\epsilon $$ϵ-subgradient methods for composite convex optimization problems," Journal of Global Optimization, Springer, vol. 75(4), pages 1029-1060, December.
    4. Xin He & Nan-jing Huang & Xue-song Li, 2022. "Modified Projection Methods for Solving Multi-valued Variational Inequality without Monotonicity," Networks and Spatial Economics, Springer, vol. 22(2), pages 361-377, June.

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