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A Redistributed Bundle Algorithm for Generalized Variational Inequality Problems in Hilbert Spaces

Author

Listed:
  • Jie Shen

    (School of Mathematics, Liaoning Normal University, Dalian 116029, P. R. China)

  • Ya-Li Gao

    (School of Mathematics, Liaoning Normal University, Dalian 116029, P. R. China)

  • Fang-Fang Guo

    (School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China)

  • Rui Zhao

    (School of Mathematics, Liaoning Normal University, Dalian 116029, P. R. China)

Abstract

Based on the redistributed technique of bundle methods and the auxiliary problem principle, we present a redistributed bundle method for solving a generalized variational inequality problem which consists of finding a zero point of the sum of two multivalued operators. The considered problem involves a nonsmooth nonconvex function which is difficult to approximate by workable functions. By imitating the properties of lower-C2 functions, we consider approximating the local convexification of the nonconvex function, and the local convexification parameter is modified dynamically in order to make the augmented function produce nonnegative linearization errors. The convergence of the proposed algorithm is discussed when the sequence of stepsizes converges to zero, any weak limit point of the sequence of serious steps xk is a solution of problem (P) under some conditions. The presented method is the generalization of the convex bundle method [Salmon, G, JJ Strodiot and VH Nguyen (2004). A bundle method for solving variational inequalities. SIAM Journal on Optimization, 14(3), 869–893].

Suggested Citation

  • Jie Shen & Ya-Li Gao & Fang-Fang Guo & Rui Zhao, 2018. "A Redistributed Bundle Algorithm for Generalized Variational Inequality Problems in Hilbert Spaces," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(04), pages 1-18, August.
    • Handle: RePEc:wsi:apjorx:v:35:y:2018:i:04:n:s0217595918500197
    DOI: 10.1142/S0217595918500197
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    References listed on IDEAS

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    1. Correa Romar, 2014. "Mathematical Foci," Mathematical Economics Letters, De Gruyter, vol. 2(1-2), pages 1-7, August.
    2. T. T. Hue & J. J. Strodiot & V. H. Nguyen, 2004. "Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(1), pages 119-145, April.
    3. G. Salmon & V. H. Nguyen & J. J. Strodiot, 2000. "Coupling the Auxiliary Problem Principle and Epiconvergence Theory to Solve General Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 629-657, March.
    4. W. Hare & C. Sagastizábal & M. Solodov, 2016. "A proximal bundle method for nonsmooth nonconvex functions with inexact information," Computational Optimization and Applications, Springer, vol. 63(1), pages 1-28, January.
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