Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces. (2nd October 2020)

• Record Type: Journal Article
• Authors: Hieu, Dang Van; Cho, Yeol Je; Xiao, Yi-bin; Kumam, Poom
• Is Part Of: Optimization. Volume 69:Number 10(2020)
• Page Start: 2279
• Page End: 2304
• Publication Date: 2020-10-02
• Subjects: Variational inequality problem -- pseudomonotone operator -- projection method -- relaxed extragradient algorithm -- Lipschitz condition; 65Y05 -- 65K15 -- 68W10 -- 47H05 -- 47H10; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2019.1683554
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : In this paper, we introduce a new algorithm for solving a variational inequality problem in a Hilbert space. The algorithm originates from an explicit discretization of a dynamical system in time. We establish the convergence of the algorithm for a class of non-monotone and Lipschitz continuous operators, provided by the sequentially weak-to-weak continuity of cost operators. The rate of convergence of the algorithm is also proved under some standard hypotheses. Moreover, the new algorithm uses variable step-sizes which are updated at each iteration by a cheap computation without linesearch. This step-size rule allows the resulting algorithm to work more easily without the prior knowledge of Lipschitz constant of operator. Also, it is particularly interesting in the case where the Lipschitz constant is unknown or difficult to approximate. Several numerical experiments are implemented to illustrate the theoretical results and also to compare with existing algorithms.