Some pivot schemes for the linear complementarity problem

Abstract

We are concerned with solutions to the LCP:

$$ \begin{gathered} w = q + {\rm M}z; w, z \geqslant 0; w_i z_i = 0; i = 1, 2. \ldots , n \hfill \\ (M is n by n; q a column). \hfill \\ \end{gathered} $$

With given column f≥0 such that for some z ₀>0, one has f+z ₀ q>0, we consider sequences of AC-pivots (“almost complementary” pivots—i.e., retaining w _i z _i =0; i=1,2,…,n) on the system:

$$w = (q + z_0 f) + Mz.$$

Mainly we are considering M such that when q admits non-negative (feasible) solutions to w=q+Mz, the sequence of AC-pivots from (2) which retains (2)-feasibility yields a solution to (1) (i.e., terminates in z ₀=0, rather than in a “z ₀-complementary ray”).

In this paper some variations of that “AC-algorithm” are given. In one version, supposing M as above, one visualizes n “sub-problems”; namely the LCP defined by the pth leading principal sub-matrix of M; for p=1,2,…,n. One finds a resolution (i.e., a sub-problem solution or termination in a ray) of the pth sub-problem by AC-pivoting, with the first p components of f fixed at appropriate values. Then, varying the (p+1)st component of f, so as to get “(p+1)-feasibility”, one proceeds to the resolution of the (p+1)st sub-problem (with “smooth transition”). Initially, p=1.

Exploiting “index theory” as initiated by Shapley, it is shown that for M as identified above, “level” p=n terminates in a solution to (1) when q+Mz=w is feasible. The same results apply for other variations; namely when M is ‘decomposed’ in more general ways. In these variations, schemas may repeat, but only at later “levels” (values of p).