A modified subgradient extragradient algorithm-type for solving quasimonotone variational inequality problems with applications

Abstract

In this article, we introduce an inertial-type algorithm that combines the extragradient subgradient method, the projection contraction method, and the viscosity method. The proposed method is used for solving quasimonotone variational inequality problems in infinite dimensional real Hilbert spaces such that it does not depend on the Lipschitz constant of the cost operator. Further, we prove the strong convergence results of the new algorithm. Our strong convergence results are achieved without imposing strict conditions on the control parameters and inertial factor of our algorithm. We utilize our algorithm to solve some problems in applied sciences and engineering such as image restoration and optimal control. Some numerical experiments are carried out to support our theoretical results. Our numerical illustrations show that our new method is more efficient than many existing methods.

On Bilevel Monotone Inclusion and Variational Inequality Problems

“…We denote by V(K, F) the solution set of the VIP (1). Problem (1) has a wide range of applications; several methods for solving this problem have been developed by many researchers (see [1][2][3] and the references in them).…”

“…However, in machine learning and CT reconstruction, strong convergence is more desirable in infinite dimensional spaces [12]. Therefore, it is necessary to modify (3), such that it can achieve strong convergence in real Hilbert spaces. In the last two decades, so many modifications of the forward-backward method have been constructed to obtain strong convergence results in real Hilbert spaces; see [11,12,15,16] and the references in them.…”

“…The idea of including inertial terms in iterative methods for solving optimization problems was initiated by Polyak [17] and it has been confirmed by numerous authors that the inclusion of an inertial term in a method acts as a boost to the convergence speed of the method. A common feature of the inertial-type algorithm is that the next iteration depends on the combination of two previous iterates; for more details, see [3,18,19]. Many inertial-type algorithms have been studied and numerical tests have demonstrated that the inertial effects on these methods greatly improve their performances; see [1,3,20].…”

“…A common feature of the inertial-type algorithm is that the next iteration depends on the combination of two previous iterates; for more details, see [3,18,19]. Many inertial-type algorithms have been studied and numerical tests have demonstrated that the inertial effects on these methods greatly improve their performances; see [1,3,20]. Recently, Lorenz and Pock [17] introduced and studied the following inertial FBA to solve the MIP (2):…”