A generalization of the equivalence relations between modulus-based and projected splitting methods

It is known that modulus-based matrix splitting methods for linear complementarity problems can be equivalently written as projected splitting methods. In this paper, we generalize this result to methods for weakly nonlinear horizontal linear complementarity problems. The presented results include, as special cases, the equivalence between methods for horizontal linear complementarity problems, nonlinear complementarity problems and linear complementarity problems. We prove the equivalence between different solution methods both in the general case and when special splittings (such as Jacobi, Gauss-Seidel and SOR) are used. Furthermore, we discuss how the equivalence can foster the formulation and the analysis of new solution methods. In this regard, we here introduce the accelerated modulus-based matrix splitting methods for weakly nonlinear horizontal linear complementarity problems. Finally, we perform a numerical evaluation of the equivalence. In this context, the efficiency of the methods and the role of the parameter γ of the modulus-based methods are also discussed.