A Jacobian smoothing method for box constrained variational inequality problems (Q1765876)

The authors propose a new Jacobian smoothing method such that it becomes well defined for general box constrained variational inequality problems: Find a vector \(x^*\in X\) such that: \(F(x^*)^T(x-x^*)\geq 0\) for all \(x\in X= \{x\in \mathbb R^n\mid \ell\leq x\leq u\}\), where \(\ell_i\in \mathbb R\cup\{-\infty\}\), \(u_i\in \mathbb R\cup \{+\infty\}\) and \(u_i >\ell_i\), \(i = 1,2\dots n\). Further, if \(X=\mathbb R^n_+\), this problem reduces to the nonlinear complementarity problem: Find \(x\in \mathbb R^n\) such that \(x\geq 0\), \(F(x)\geq 0\), \(x^TF(x)= 0\). By the Fischer-Burmeister function \(h: \mathbb R^2\to \mathbb R\), \(h(a, b) =\sqrt{a^2+ b^2} -a-b\), this problem is equivalent to the nonlinear system \[ H(z)=\left[\begin{aligned} & h(x_1-\ell_1,F_1(x)+y_1)\\ &\vdots \\ & h(x_n-\ell_n,\,F_n(x)+y,)\\ & h(u_1-x_1,y_1)\\ & \vdots \\ & h(u_n -x_n,y_n)\end{aligned}\right]=0\tag{1} \] where \(z=(x^Tmy^T)^T\in \mathbb R^{2n}\), the nonsmoothing mapping \(H: \mathbb R^{2n}\to \mathbb R^{2n}\). Main result: The globalization strategy for the proposed algorithm is based on the natural merit function \(\theta:\mathbb R^{2n}\to \mathbb R_+\) given by: \(\theta(z)=\frac12\| H(z)\|^2\), where \(z=(x^T,y^T)\in \mathbb R^{2n}\). The corresponding smooth operator \(H_\mu: \mathbb R^{2n}\to \mathbb R^{2n}\) is defined by (1) with \(h=h_\mu :\mathbb R^2\to \mathbb R\) (the smooth approximation). Here \[ h_\mu(a,b)=\begin{cases} \sqrt{a^2+b^2}-a-b, \sqrt{a^2+b^2}>\mu\\ \frac{a(a-2\mu)}{2\mu}+\frac{b(b-2\mu)}{2\mu}+\frac{\mu}{2},\;\sqrt{a^2+b^2}\leq \mu\end{cases} \] of the Fischer-Burmeister function. The Jacobian consistency property and the Jacobian smoothing idea are discussed. The algorithm is proposed in detail (precise proofs are devoted to proving global superlinear convergence of the algorithm). Finally, some numerical experiments for the proposed algorithm are presented.