A modified standard embedding for linear complementarity problems

Abstract

We propose a modified standard embedding for solving the linear complementarity problem (LCP). This embedding is a special one-parametric optimization problem $P(t), t\in [0,1]$. Under the conditions (A3) (the Mangasarian-Fromovitz Constraint Qualification is satisfied for the feasible set $M(t)$ depending on the parameter $t$), (A4) ($P(t)$ is Jongen-Jonker- Twilt regular) and two technical assumptions (A1) and (A2) there exists a path in the set of stationary points connecting the chosen starting point for $P(0)$ with a certain point for $P(1)$, and this point is a solution of the (LCP). This path may include types of singularities, namely points of Type 2 and Type 3 in the class of Jongen-Jonker-Twilt for $t\in [0,1)$. We can follow this path by using pathfollowing procedures (contained in the program package PAFO). In case that the condition (A3) is not satisfied, also points of Type 4 and 5 may appear. The assumption (A4) will be justified by a theorem. Illustrative examples are presented.