Self-adaptive inertial single projection methods for variational inequalities involving non-Lipschitz and Lipschitz operators with their applications to optimal control problems

Abstract

In this paper, four accelerated subgradient extragradient methods are proposed to solve the variational inequality problem with a pseudo-monotone operator in real Hilbert spaces. These iterative schemes employ two new adaptive stepsize strategies that are significant when the Lipschitz constant of the mapping involved is unknown. Strong convergence theorems for the proposed algorithms are established under the condition that the operators are Lipschitz continuous and non-Lipschitz continuous. Numerical experiments on finite- and infinite-dimensional spaces and applications in optimal control problems are reported to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.