A modified penalty embedding for linear complementarity problems

We propose a modified penalty embedding for solving complementarity problem (LCP). This embedding is a special one parametric optimization problem P(t), t∈[0,1]. Under the condition (A3) (a modified Enlarged Mangasarian Fromovitz Constrait Qualification), (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 for (LCP). The path may include types of singularities, namely points of Type 2, Type 3 and Type 4 in the class of Jongen–Jonker–Twilt. We can follow this path by using pathfollowing procedures (program package PAFO) only. We do not have any assumption with respect to the matrix B in the description of the (LCP). The assumption (A4) will justified by two theorems. An illustrative example shows that points of Type 2 and 3 could appear.