A new modified one-step smoothing Newton method for solving the general mixed complementarity problem
The mixed complementarity problem to find a vector \(x \in X \subset \mathbb R^n\) such that \( F(x)^T (y-x) \geq0\), \(\forall y\in X\), where \(X=\prod^n_{i=1} [l_i, u_i]\), \(-\infty \leq l_i < u_i < + \infty\), \(i = 1,2, \dots, n\) is analyzed. Furthermore, if \(X=\mathbb R^n_+\), the above problem reduces to the nonlinear complementarity problem, which is to find \(x \in\mathbb R^n\), such that \(x\geq0\), \(F(x) \geq 0\), \(x^TF(x)=0\). This corresponds to the case that \(l_i=0\) and \(u_i= +\infty\) for all \(i=1,2,\dots, n\), and for \(l_i=-\infty\), \(u_i=+\infty\), \(i=1,2,\dots, n\). In this case, the mixed complementarity problem reduces to the nonlinear equation \(F(x)=0\) and can be reformulated as system of nonsmooth equations: \(\Phi(x)=0\) where \(\Phi:\mathbb R^n \rightarrow\mathbb R^n\) is a semismooth function and its square function, \(\Psi (x):=\frac{1}{2} \|\Phi(x)\|^2\), is continuously differentiable. Main result: A detailed description of the authors modified one-step smoothing Newton method (based on the well-known Chen-Harker-Kanzov-Smale smooth function) and the state of some of its elementary properties is given. It is shown that the presented algorithm is well-defined for an arbitrary mixed complementarity problem. The authors' algorithm consists of two parts: a Newton step, which is in a similar (but slightly different) way as this is done by some smoothing Newton method, and a gradient step. The two parts adopt different merit functions. The global convergence and local superlinear convergence of the algorithm are discussed.
modified one-step smoothing Newton method
Chen-Harker-Kanzov-Smale smooth function
scientific article; zbMATH DE number 5711778
scientific article