We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.

In recent years, much attention has been given to study the variational inequalities and related problems on the Riemannian manifold and the Hadamard manifold. This framework is a useful for the developments of various fields. Several ideas and techniques from the Euclidean space have been extended and generalized to this nonlinear framework. The Hadamard manifolds are examples of hyperbolic spaces and geodesics; see [

We now recall some fundamental and basic concepts needed for a reading of this paper. These results and concepts can be found in the books on the Riemannian geometry [

Let

Let

A Riemannian manifold is complete, if for any

Let

A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a

We also recall the following well-known results, which are essential for our work.

Let

So from now on, when referring to the geodesic joining two points, we mean the unique minimal normalized one. Lemma

Let

Let

From the law of cosines in inequality (

Let

For any

If

Given the sequences

A subset

A real-valued function

The subdifferential of a function

The existence of subgradients for convex functions is guaranteed by the following proposition; see [

Let

For a given single-valued vector field

We remark that if the function

An operator

We now use the auxiliary principle technique of Glowinski et al. [

For a given

For a given

If

For a given

If

For a given

We would like to mention that Algorithm

For a given

In a similar way, one can obtain several iterative methods for solving the variational inequalities on the Hadamard manifold.

We now consider the convergence analysis of Algorithm

Let

Let

Let

Let

We have used the auxiliary principle technique to suggest and analyzed a proximal point iterative method for solving the mixed quasi-variational inequalities on the Hadamard manifolds. Some special cases are also discussed. Convergence analysis of the new proximal point method is proved under weaker conditions. Results obtained in this paper may stimulate further research in this area. The implementation of the new method and its comparison with other methods is an open problem. The ideas and techniques of this paper may be extended for other related optimization problems.

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, the COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities.