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A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces

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  • J. Y. Bello Cruz

    (Federal University of Goiás)

  • R. Díaz Millán

    (Federal Institute of Education, Science and Technology)

Abstract

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:65:y:2016:i:3:d:10.1007_s10898-015-0397-x
    DOI: 10.1007/s10898-015-0397-x
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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.
    3. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, December.
    4. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    5. J. Bello Cruz & W. Oliveira, 2014. "Level bundle-like algorithms for convex optimization," Journal of Global Optimization, Springer, vol. 59(4), pages 787-809, August.
    6. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    7. Bello Cruz, J.Y. & Iusem, A.N., 2015. "Full convergence of an approximate projection method for nonsmooth variational inequalities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 114(C), pages 2-13.
    8. 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.
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    Cited by:

    1. 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.

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