New error bounds for the extended vertical LCP

Abstract

In this paper, by making use of this fact that for $a_{j}, b_{j}\in \mathbb{R}$, $j=1,2,\ldots,n$, there are $\lambda_{j}\in [0,1]$ with $\sum_{j=1}^{n}\lambda_{j}=1$ such that \[ \min_{1\leq j\leq n}\{a_{j}\}-\min_{1\leq j\leq n}\{b_{j}\}=\sum_{j=1}^{n}\lambda_{j}(a_{j}-b_{j}), \] some new error bounds of the extended vertical LCP under the row $\mathcal{W}$-property are obtained, which cover the error bounds in [Math. Program., 106 (2006) 513-525] and [Comput. Optim. Appl., 42 (2009) 335-352]. Not only that, these new error bounds skillfully avoid the inconvenience caused by the row rearrangement technique for error bounds to achieve the goal of reducing the computation workload, which was introduced in the latter paper mentioned above. Besides, with respect to the row $\mathcal{W}$-property, two new sufficient and necessary conditions are obtained.