A strongly polynomial rounding procedure yielding a maximally complementary solution for P(*)(k) linear complementarity problems

We deal with linear complementarity problems (LCPs) with P(*)(kappa) matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter mu, as usual. Our elementary proof reproduces the known result that the variables on or close to the central path fall apart in three classes in which these variables are O(1), O(mu), and O(root mu), respectively. The constants hidden in these bounds are expressed in or bounded by the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and which is close enough to the central path, the rounding procedure produces a maximally complementary solution in at most O(n(3)) arithmetic operations. The result implies that interior point methods (IPMs) not only converge to a complementary solution of P(*)(kappa) LCPs, but, when furnished with our rounding procedure, they can also produce a maximally complementary (exact) solution in polynomial time.