Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems

• Published in: Numerical algorithms Vol. 89; no. 4; pp. 1695 - 1721
• Main Authors: Linh, Ha Manh, Reich, Simeon, Thong, Duong Viet, Dung, Vu Tien, Lan, Nguyen Phuong
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2022; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Convergence; Hilbert space; Iterative methods; Numeric Computing; Numerical Analysis; Operators (mathematics); Original Paper; Theory of Computation; Fixed point problem; Strong convergence; 65K15; 47J20; Contraction and projection method; 90C25; Convergence rate; Variational inequality problem; 47H09; Demicontractive mapping; Inertial method; Relaxed inertial gradient method
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-021-01169-8

We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the Lipschitz constant of the variational inequality operator and that it only computes one projection onto the feasible set per iteration. In addition, we do not need the sequential weak continuity of the variational inequality operator in order to establish our strong convergence theorem. Next, we also obtain an
R
-linear convergence rate for a related relaxed inertial gradient method under strong pseudo-monotonicity and Lipschitz continuity assumptions on the variational inequality operator. Finally, we present several numerical examples which illustrate the performance and the effectiveness of our algorithm.