A non-interior-point smoothing method for variational inequality problem (Q966082)

In this very useful paper, the authors study the variational inequality problem: find a vector \(x^*\in X \) such that for all \(x\in X (x-x^*)^Tf(x^*)\geq 0\), where \(f: X\rightarrow\mathbb R^n\) is continuously differentiable function. It is an important class of optimization problem and contains the linear (quadratic) programming, the system of equations of the box constrained variational inequality problem. The important case of the proposed problem is the nonlinear complementarity problem, which is to find \(x \in\mathbb R^n_+\) such that \(f^T(x)x=0\), \(f(x)\in\mathbb R^n_+\), where \(f\) is a mapping from \(\mathbb R^n_+\) to \(\mathbb R^n\). Main result: A non-interior-point smoothing algorithm for the variational inequality problem is presented and that the algorithm is well-defined is proved. If \(f\) is a \(P_0\) function (when for all pairs of distinct vectors \(x,y\in\mathbb R^n\), there exists \(i\in \{1,2,\dots,n\}\) such that \(x_i \neq y_i\) and \((x_i - y_i)^T (f_i(x)-f_i(y)) \geq 0),\) the global convergence and locally quadratic convergence under the nonsingularity assumption is proved. Further the locally quadratic convergence in the absence of the strict complementarity conditions is shown. Finally, the experimental results indicate authors' algorithm performs well.