A Krylov Subspace Method for Large-scale Second-order Cone Linear Complementarity

In this paper, we first show that the second-order cone linear complementarity problem (SOCLCP) can be solved by finding a positive zero s∗ ∈ R of a particular rational function h(s), and we then propose a Krylov subspace method to reduce h(s) to h (s) as in the model reduction. The zero s∗ of h(s) can be accurately approximated by that of h (s) = 0, which itself can be cast as a small eigenvalue problem. The new method is made possible by a complete characterization of the curve of h(s), and it has several advantages over the bisection-Newton (BN) iteration recently proposed by [L.-H. Zhang and W. H. Yang, Math. Comp., 83 (2013), pp. 1701–1720] and shown to be very efficient for smallto medium-size problems. The method is tested and compared against the BN iteration and two other state-of-the-art packages: SDPT3 and SeDuMi. Our numerical results show that the method is very efficient for both small to medium dense problems and large-scale ones.