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تسجيل مستخدم جديدAn inexact Douglas-Rachford splitting method for solving absolute value equations
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The last two decades witnessed the increasing of the interests on the absolute value equations (AVE) of finding $xinmathbb{R}^n$ such that $Ax-|x|-b=0$, where $Ain mathbb{R}^{ntimes n}$ and $bin mathbb{R}^n$. In this paper, we pay our attention on designing efficient algorithms. To this end, we reformulate AVE to a generalized linear complementarity problem (GLCP), which, among the equivalent forms, is the most economical one in the sense that it does not increase the dimension of the variables. For solving the GLCP, we propose an inexact Douglas-Rachford splitting method which can adopt a relative error tolerance. As a consequence, in the inner iteration processes, we can employ the LSQR method ([C.C. Paige and M.A. Saunders, ACM Trans. Mathe. Softw. (TOMS), 8 (1982), pp. 43--71]) to find a qualified approximate solution for each subproblem, which makes the cost per iteration very low. We prove the convergence of the algorithm and establish its global linear rate of convergence. Comparing results with the popular algorithms such as the exact generalized Newton method [O.L. Mangasarian, Optim. Lett., 1 (2007), pp. 3--8], the inexact semi-smooth Newton method [J.Y.B. Cruz, O.P. Ferreira and L.F. Prudente, Comput. Optim. Appl., 65 (2016), pp. 93--108] and the exact SOR-like method [Y.-F. Ke and C.-F. Ma, Appl. Math. Comput., 311 (2017), pp. 195--202] are reported, which indicate that the proposed algorithm is very promising. Moreover, our method also extends the range of numerically solvable of the AVE; that is, it can deal with not only the case that $|A^{-1}|<1$, the commonly used in those existing literature, but also the case where $|A^{-1}|=1$.
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In Wang et al. (J. Optim. Theory Appl., textbf{181}: 216--230, 2019), a class of effective modified Newton-tpye (MN) iteration methods are proposed for solving the generalized absolute value equations (GAVE) and it has been found that the MN iteratio
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nd Ma ([{em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $|T_ u(omega)|_2$ with $$T_ u(omega) = left(begin{array}{cc} |1-omega| & omega^2 u |1-omega| & |1-omega| +omega^2 u end{array}right)$$ and the approximate optimal parameter which minimizes $eta_{ u}(omega) =max{|1-omega|, uomega^2}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ u$ is, the smaller convergent region of the iteration parameter $omega$ is. Numerical results are presented to 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 ([{em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.
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