A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

The objective of this work is to design a new iterative method based on Armijo's type-modified extragradient method for solving the inclusion problem (A + B)(-1)(0), where A is a maximal monotone vector field and B is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the convergence rate. A convergence theorem is established for the proposed extragradient method, significantly improving existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field's monotonicity can be removed.