Golden Ratio Algorithm with Inertia for Non-Lipschitz Variational Inequalities

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation steps. In order to speed up the convergence of projection and contraction methods with inertial steps for solving variational inequalities, we propose a new approach that combines double inertial extrapolation steps, the modified Mann-type projection and contraction method, and the line-search rule, which is based on the golden ratio (5+1)/2. We demonstrate the efficiency, robustness, and stability of the suggested algorithm with numerical examples.

In this paper, we combine the golden ratio method with the projection algorithm to obtain the golden ratio projection algorithm. In order to get better convergence speed, we also propose an alternating golden ratio projection algorithm. Unlike ordinary inertial extrapolation, golden ratio method is constructed based on a convex combined structure about the entire iterative trajectory. The advantages of the proposed algorithms require only one projection onto the feasible set and do not require knowledge of the Lipschitz constant for the operator since our algorithms use self-adaptive step-sizes. The R-linear convergence results of the two algorithms are established for strongly pseudo-monotone variational inequality. Finally, we present some numerical experiments to show the efficiency and advantages of the proposed algorithms.

Backward Inertial Modification of the Bregman Golden Ratio Algorithm for Solving Variational Inequalities

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

An improved subgradient extragradient self-adaptive algorithm based on the golden ratio technique for variational inequality problems in Banach spaces