Penalty function method for a variational inequality on Hadamard manifolds

The present paper deals with penalty function method for K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}$$\end{document}-constrained variational inequality problem on Hadamard manifolds. It transforms the constrained problem into an unconstrained problem and named it a penalized variational inequality problem. We establish sufficient condition under the assumption of coercivity for solving both problems and illustrate by a non-trivial example. It is shown that the sequence of a solution to the penalized variational inequality problem has at least one limit point within the feasible region. Moreover, it is also observed that any limit point of the sequence of a solution to the penalized variational inequality problem also is a solution to the original problem.