Solving strongly monotone linear complementarity problems

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q V we consider the general linear complementarity problem LCP(L, K, q) on a proper cone K E; V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP(L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example. © 2013 World Scientific Publishing Company.