Modulus-based matrix splitting algorithms for generalized complex-valued horizontal linear complementarity problems

Abstract

In this paper, we introduce the complex-valued horizontal linear complementarity problem (CHLCP), we provide two equivalent real-valued reformulations, and study modulus-based matrix splitting algorithms for solving the CHLCP. This latter point is motivated by the recent introduction of modulus-based matrix splitting methods for (non-horizontal) complex linear complementarity problems (CLCPs), which we generalize. We study the convergence of the proposed algorithms. Whenever possible, we seek convergence conditions that are directly based on the form of the real and imaginary parts of the matrices of the CHLCP in its complex form. This makes the convergence easier to evaluate than in existing convergence analyses. Finally, we study the numerical properties of the proposed algorithms by solving several CHLCPs. In this context, we also revisit results on the CLCP under the larger CHLCP framework, providing new numerical insights on the efficiency of existing algorithms for the CLCP.