A QP-free constrained Newton-type method for variational inequality problems (Q1295959)

The authors consider the following variational inequality problem: find a vector \(x^*\in X\) such that \[ F(x^*)^T*x- x^*)\geq 0\quad\text{for all }x\in X, \] where \(F: \mathbb{R}^n\to \mathbb{R}^n\) is a continuously differentiable function and \(X\) is the set of all vectors \(x\in \mathbb{R}^n\) satisfying the constraints \(g(x)\geq 0\) and \(h(x)= 0\) given by the twice continuously differentiable functions \(g: \mathbb{R}^n\to \mathbb{R}^m\) and \(h: \mathbb{R}^n\to \mathbb{R}^p\). Instead of solving this problem directly, they try to find a Karush-Kuhn-Tucker point of it. To this end they formulate a constrained minimization problem, whose stationary points are Karush-Kuhn-Tucker points under reasonable conditions, and describe an algorithm which yields such stationary points. This new algorithm does not use quadratic programs. At each iteration only one linear system of equations has to be solved.