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

Jie Shen, Ya-Li Gao (), Fang-Fang Guo () and Rui Zhao ()
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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

Asia-Pacific Journal of Operational Research (APJOR), 2018, vol. 35, issue 04, 1-18

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].

Keywords: Redistributed bundle method; subgradient; generalized variational inequality problem; zero point of operator; auxiliary problem principle; gap function (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1142/S0217595918500197

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