Axel G.R. Turnquist , Xiaoming John Zhang

Olawale Kazeem Oyewole

Thursday, September 17, 2026 10:00 AM - 11:00 AM

A3-1-301

Zoom 242 742 6089 (BIMSA)

This talk presents a comprehensive overview of the Variational Inequality Problem (VIP), from its origins in the works of Fichera (1963) and Stampacchia (1968) to modern solution methods and applications. We begin with the foundational tools: fixed-point theorems and the metric projection operator, which yield the classical characterization of variational inequalities as fixed-point problems. After formulating the VIP and examining its connections to complementarity and minimization problems, we discuss existence and uniqueness results and the associated monotonicity conditions. The presentation surveys the frameworks in which the problem has been studied, from Euclidean and Hilbert spaces to reflexive Banach spaces and Hadamard manifolds, before reviewing iterative solution methods, including extragradient, subgradient extragradient, Popov, and Tseng variants. We then present our contributions: inertial and self-adaptive algorithms with strong convergence guarantee in Banach spaces, a schematic approximation method on Hadamard manifolds, and an inertial Tseng extragradient method for stochastic variational inequalities. Finally, we discuss image processing as an application to our variational inequality methods.