The Complexity of High-Order Interior-Point Methods for Solving Sufficient Complementarity Problems

Abstract

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 Cottle, Pang and Venkateswaran [1]. 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 we report on some recent results on the complexity of these methods. We outline a proof that these methods need at most $$\mathcal{O}\left( {(1 + \kappa )\sqrt {n|\log \varepsilon |} } \right)$$ steps to compute an ε-solution, if the problem has strictly interior points. Here, к is the sufficiency parameter of the complementarity problem.