Mathematical Programming

, Volume 53, Issue 1–3, pp 99–110

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

• Masao Fukushima
Article

## Abstract

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 an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show 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. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.

## Key words

Asymmetric variational inequality problem equivalent differentiable optimization problem descent methods nonlinear complementarity problem nonlinear equations monotone mapping

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