The complexity of high-order predictor-corrector methods for solving sufficient linear complementarity problems

Recently the authors of this paper and S. Mizuno described a class of infeasible-interior-point methods for solving linear complementarity problems that are sufficient in the sense of R.W. Cottle, J.-S. Pang and V. Venkateswaran (1989) Sufficient matrices and the linear complementarity problem, Linear Algebra Appl. 114/115, 231-249. It was shown that these methods converge superlinearly with an arbitrarily high order even for degenerate problems or problems without strictly complementary solution. In this paper the complexity of these methods is investigated. It is shown that all these methods, if started appropriately, need O((1 + kappa)(2)n\log epsilon\) predictor-corrector steps to find an epsilon-solution, and only O((1+ kappa)root n\log epsilon\) steps, if the problem has strictly interior points. Here, kappa is the sufficiency parameter of the complementarity problem.