Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations
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The absolute value equations (AVE) Ax - |x| - b = 0 is of interest of the optimization community. Recently, the SOR-like iteration method has been developed (Ke and Ma [ Appl. Math. Comput., 311:195–202, 2017]) and shown to be efficient for numerically solving the AVE with ν=A^-1_2<1 (Ke and Ma [ Appl. Math. Comput., 311:195–202, 2017]; Guo, Wu and Li [ Appl. Math. Lett., 97:107–113, 2019]). Since the SOR-like iteration method is one-parameter-dependent, it is an important problem to determine the optimal iteration parameter. In this paper, we revisit the convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([ Appl. Math. Comput., 311:195–202, 2017]). Furthermore, we explore the optimal parameter which minimizes T(ω)_2 and the approximate optimal parameter which minimizes η=max{|1-ω|,νω^2}. The optimal and approximate optimal parameters are iteration-independent. Numerical results demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([ Appl. Math. Lett., 97:107–113, 2019]).
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