Convergence of the projection and contraction methods for solving bilevel variational inequality problems

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 9; pp. 10867 - 10885
• Main Authors: Thang, Tran Van, Anh, Pham Ngoc, Truong, Nguyen Duc
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01-06-2023
• Subjects: Algorithms; bilevel problem; contraction; Convergence; Iterative methods; Lipschitz continuous; Mapping; monotone; projection method; variational inequality problem
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9156

In this work, we analyze some convergent properties of a projection and contraction algorithm for solving a variational inequality problem, where the feasible domain is the solution set of an affine variational inequality problem. We prove that, for solving the problem where the second cost mapping is affine and not necessary for monotone properties, any iterative sequence generated by the algorithm converges to a unique solution provided that the first cost mapping is strongly monotone and Lipschitz continuous. Computational errors of the algorithm are showed. Finally, some preliminary numerical experiences and comparisons are also reported.