A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems

Abstract

We consider the standard linear complementarity problem (LCP): Find (x, y) ∈ R ²ⁿ such that y = M x + q, (x, y) ≥ 0 and x _i y _i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P ₀ LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in $O\left( {\frac{{\gamma ^{ - 6} n}}{{\varepsilon ^6 }}\log \frac{{\gamma ^{ - 2} n}}{{\varepsilon ^2 }}} \right)$ Newton iterations where ${\bar \gamma }$ is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods.