A modified multivariate spectral gradient algorithm for solving absolute value equations

Abstract In this paper, a modified multivariate spectral gradient algorithm is proposed for solving the absolute value equations  A x − | x | = b , where  A ∈ R n × n , b ∈ R n and  x ∈ R n . Some properties of the absolute value equations are analyzed, and the global convergence of the proposed algorithm based on some appropriate assumptions is discussed. Several examples are given to illustrate the effectiveness and competitiveness of our algorithm.

[1]  O. Mangasarian,et al.  Absolute value equations , 2006 .

[2]  Davod Khojasteh Salkuyeh,et al.  The Picard–HSS iteration method for absolute value equations , 2014, Optim. Lett..

[3]  Le Han,et al.  Multivariate spectral gradient method for unconstrained optimization , 2008, Appl. Math. Comput..

[4]  Lou Caccetta,et al.  A globally and quadratically convergent method for absolute value equations , 2011, Comput. Optim. Appl..

[5]  Olvi L. Mangasarian,et al.  Absolute value equation solution via concave minimization , 2006, Optim. Lett..

[6]  Aixiang Wang,et al.  Interval algorithm for absolute value equations , 2011 .

[7]  Le Han,et al.  Hierarchical-Multivariate Spectral Gradient Method and Its Applications in Nonrigid Registration of Medical Image , 2009, 2009 First International Conference on Information Science and Engineering.

[8]  Muhammad Aslam Noor,et al.  On an iterative method for solving absolute value equations , 2012, Optim. Lett..

[9]  Hao Liu,et al.  A verification method for enclosing solutions of absolute value equations , 2013 .

[10]  Yunhai Xiao,et al.  Spectral gradient projection method for monotone nonlinear equations with convex constraints , 2009 .

[11]  Olvi L. Mangasarian,et al.  A generalized Newton method for absolute value equations , 2009, Optim. Lett..

[12]  Yi Qin,et al.  A multivariate spectral projected gradient method for bound constrained optimization , 2011, J. Comput. Appl. Math..

[13]  Olvi L. Mangasarian.  A hybrid algorithm for solving the absolute value equation , 2015, Optim. Lett..

[14]  Jiri Rohn,et al.  A theorem of the alternatives for the equation |Ax| − |B||x| = b , 2004, Optimization Letters.

[15]  Farhad Khaksar Haghani,et al.  On Generalized Traub’s Method for Absolute Value Equations , 2015, J. Optim. Theory Appl..

[16]  Jiri Rohn,et al.  An algorithm for solving the absolute value equation , 2009 .

[17]  Peng Guo,et al.  On the SOR-like iteration method for solving absolute value equations , 2019, Appl. Math. Lett..

[18]  Jiri Rohn,et al.  An iterative method for solving absolute value equations and sufficient conditions for unique solvability , 2014, Optim. Lett..

[19]  Qiong Zhang,et al.  A generalized Newton method for absolute value equations associated with second order cones , 2011, J. Comput. Appl. Math..

[20]  Orizon Pereira Ferreira,et al.  On the global convergence of the inexact semi-smooth Newton method for absolute value equation , 2015, Computational Optimization and Applications.

[21]  Jianhua Ma,et al.  Multivariate spectral gradient projection methodfor nonlinear monotone equations with convex constraints , 2012 .