Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives a affirmative answer to this question. It provides a new optimization problem formulation of the variational inequality problem, and shows that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. The paper also shows that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. It discusses descent methods, for solving the equivalent optimization problem and comments on systems of nonlinear equations and nonlinear complementarity problems.