Further Notes on the Existence of Search

Following the work of Shida, Shindoh, and Kojima and of Monteiro and Zanjacomo, we study the issue of existence and uniqueness of search directions in interior-point methods for the monotone LCP over the space of n n symmetric block-diagonal matrices, with particular focus on the direction of Alizadeh, Haeberly, and Overton (AHO). We give an ana-log of a result of Shida et al., characterizing existence and uniqueness in terms of a certain composite linear mapping being strictly monotone. This characterization yields a short and uniied derivation of existing suf-cient conditions for existence and uniqueness of the AHO direction. In the n = 2 case, this characterization may be expressed by two algebraic inequalities and be used to show that the \1=2" in the suucient condition of Monteiro and Zanjacomo cannot be improved beyond a certain number less than 1.