A smoothing Newton method for general nonlinear complementarity problems
The paper deals with the nonlinear complementarity problem: to find \(x \in \mathbb{R}^n_+\) such that \(F(x) \geq 0\), \(x \geq 0\), \(x^TF(x)=0\), where \(F: \mathbb{R}^n\to \mathbb{R}^n\) is a continuously differentiable function. The authors consider the system: \(H= ( e^{\mu}-1 \Phi_{\mu} (x))^T\), where \(\Phi_{\mu}\) is defined by Kanzow's smoothing approximation of the Fisher-Burmeister function (\(\sqrt{\mu^2 + x_i^2 +F_i (x)^2} -x_i- F_i(x)\)). Some properties of \(H\) , \(\Phi_\mu\) and \(\chi\) which play an important role in the analysis of the methods are discussed in the paper. For solving the above system the authors present an algorithm, which combines Newton steps and gradient steps. Two different merit functions are used: \(f(\mu,x)= \frac{ \|H(\mu,x) \|^2}{2}\) for Newton step and \(\chi (x)= \frac{\|\Phi_ \mu(x) \|^2}{2}\) for gradient step. In the Newton step \(\mu^k\) is a variable, in the gradient step \(\mu_k\) is viewed as a smoothing parameter which has to be updated. The rules for updating \(\mu\), the restriction of the step length along Newton direction assure the global convergence of the algorithm without boundedness assumption. The nonsingularity of any \(H(0,x^*)\) (\(x^*\) accumulation point of the sequence generated by the algorithm) is assumed for proving the local convergence of the algorithm. The paper includes a brief description of the implementation of the algorithm and reports numerical results for the test problems in MCPLIB and GAMSLIB libraries.
scientific article; zbMATH DE number 1565502