A two-step inertial method with a new step-size rule for variational inequalities in hilbert spaces
Jian-Wen Peng, Jun-Jie Luo, Abubakar Adamu
TL;DR
This work tackles the variational inequality problem $VI(A,C)$ in real Hilbert spaces under quasimonotonicity by introducing a two-step inertial Tseng extragradient method equipped with self-adaptive and Armijo-like step-size rules. The algorithm leverages information from two preceding iterates to enhance momentum while adaptively selecting the step-size to avoid dependence on a Lipschitz constant. The main theoretical contribution is a weak convergence proof to a dual VIP solution $S_D$ (hence to $S$ under suitable conditions) without requiring Lipschitz continuity of $A$, along with a bound-preserving update mechanism for the step-size. This approach broadens applicability to a wider class of VIPs and offers potential improvements over existing methods by reducing step-size tuning and avoiding multiple projections per iteration.
Abstract
In this paper, a two-step inertial Tseng extragradient method involving self-adaptive and Armijo-like step sizes is introduced for solving variational inequalities with a quasimonotone cost function in the setting of a real Hilbert space. Weak convergence of the sequence generated by the proposed algorithm is proved without assuming the Lipschitz condition. An interesting feature of the proposed algorithm is its ability to select the better step size between the self-adaptive and Armijo-like options at each iteration step. Moreover, removing the requirement for the Lipschitz condition on the cost function broadens the applicability of the proposed method. Finally, the algorithm accelerates and complements several existing iterative algorithms for solving variational inequalities in Hilbert spaces.