The solvability of weighted complementarity problems and a smoothing Newton algorithm under the local error bound

Abstract

The weighted Complementarity Problem (wCP) can be used for modelling a larger class of problems from science and engineering. In this paper, we first extend the solvability of the Complementarity Problems (CP) obtained by Yoshise [SIAM J. Optim. 17, 1129–1153 (2006)] to the wCP. Especially, we prove that the solution set of the wCP is bounded. We then propose a smoothing Newton algorithm to solve the wCP and prove that it is globally convergent. Moreover, we establish the local quadratic convergence of the proposed algorithm under the local error bound condition which is weaker than the nonsingularity condition used in previous smoothing Newton-type algorithms. Further, we apply the proposed algorithm to solve the weighted horizontal linear complementarity problem and report some numerical results.