Strong convergence of multi-parameter projection methods for variational inequality problems

• Published in: Mathematical modelling and analysis Vol. 27; no. 2; pp. 242 - 262
• Main Authors: Van Hieu, Dang, Dung Muu, Le, Kim Quy, Pham
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 27.04.2022
• Subjects: Analysis; Convergence; Equality; Half spaces; Hilbert space; Inequality; iterative method; Lipschitz continuity; Methods; monotonicity; Optimal control; Parameters; Regularization; variational inequality; variational inequality; monotonicity; iterative method; regularization; Lipschitz continuity
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2022.14479

In this paper, we introduce a multi-parameter projection method for solving a variational inequality problem, and establish its strong convergence in a Hilbert space under appropriate conditions. The method involves two projectionsteps with different variable stepsizes where one of them is computed explicitly on a specifically structural half-space. The proof of strong convergence of the method is based on the regularization solutions depending on parameters of the original problem. It turns out that the solution obtained by the method is the solution of a bilevel variational inequality problem whose constraint is the solution set of our considered problem. In order to support the obtained theoretical results, we perform some experiments on transportation equilibrium and optimal control problems, and also involve comparisons. Numerical results show the computational effectiveness and the fast convergence of the new method over some existing ones.