A smoothing inexact Newton method for variational inequalities with nonlinear constraints

Zhili Ge,Qin Ni,Xin Zhang

doi:10.1186/s13660-017-1433-9

Abstract

In this paper, we propose a smoothing inexact Newton method for solving variational inequalities with nonlinear constraints. Based on the smoothed Fischer-Burmeister function, the variational inequality problem is reformulated as a system of parameterized smooth equations. The corresponding linear system of each iteration is solved approximately. Under some mild conditions, we establish the global and local quadratic convergence. Some numerical results show that the method is effective.

Highlights

IntroductionWe consider the variational inequality problem (VI for abbreviation), which is to find a vector x∗ ∈ such that
We consider the variational inequality problem (VI for abbreviation), which is to find a vector x∗ ∈ such thatVI(, F) x – x∗ F x∗ ≥, ∀x ∈, ( )where is a nonempty, closed and convex subset of Rn and F is a continuous differentiable mapping from Rn into Rn
When = Rn+, VI reduces to the nonlinear complementarity problem (NCP for abbreviation) x∗ ∈ Rn+, F x∗ ∈ Rn+, x∗ F x∗ =

Summary

Introduction

We consider the variational inequality problem (VI for abbreviation), which is to find a vector x∗ ∈ such that. Based on the above symmetric perturbed Fischer-Burmeister function ( ), Chen et al [ ] proposed the first globally and superlinearly convergent smoothing Newton method. They dealt with general box constrained variational inequalities. Assume that F is continuously differentiable and strongly monotone, g is twice continuously differentiable concave, (μ∗, x∗, λ∗, z∗) in R+ × Rn × Rm × Rm is the solution of H(μ, x, λ, z) = , the rows of ∇g(x∗) are linearly independent and (λ∗, z∗) satisfies the strict complementarity condition. Since (λ∗, z∗) satisfies the strict complementarity condition, i.e., λ∗i , zi∗ are not equal to at the same time, we have that H is continuously differentiable on (μ∗, x∗, λ∗, z∗).

The global inexact algorithm and its convergence

Conclusions