Introducing three new smoothing functions: Analysis on smoothing-Newton algorithms

[1] L. Abdallah, M. Haddou, T. Migot, Solving absolute value equation using complementarity and
smoothing functions, J. Comput. Appl. Math. 327 (2018) 196–207.
[2] J.H. Alcantara, J.S. Chen, M.K. Tam, Method of alternating projections for the general absolute
value equation, J. Fixed Point Theory Appl. 25 (2023) 39.
[3] R. Ali, K. Pan, The solution of a type of absolute value equations using two new matrix splitting
iterative techniques, Port. Math. 79 (2023) 241–252.

[4] R. Ali, K. Pan, New generalized GaussSeidel iteration methods for solving absolute value equa-
tions, Math. Methods Appl. Sci. (2023) Doi: https://doi.org/10.1002/mma.9062.
[5] L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pacific
J. Math. 16 (1966) 1–3.
[6] D. Bertsekas, Nondifferentiable optimization via approximation, Math. Programming Stud.3 (1975)
1–25.
[7] L. Caccetta, B. Qu, G. Zhou, A globally and quadratically convergent method for absolute value
equations, Comput. Optim. Appl. 48 (2011) 45–58.
[8] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. 134 (2012)
71–99.
[9] C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complemen-
tarity problem, Comput. Optim. Appl. 5 (1996) 97–138.
[10] C. Chen, Y. Yang, D. Yu, D. Han, An inverse-free dynamical system for solving the absolute value
equations, Appl. Numer. Math. 168 (2021) 170–181.
[11] X. Fan, M. Zeng, L. Jiang, Smoothing homotopy method for solving second-order cone comple-
mentarity problem, Asia-Pac. J. Oper. Res. 37 (2020) 2050023.
[12] C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathe-
matics, In book Panorama of Mathematics: Pure and Applied, 658 (2016) 249–265.
[13] F. Hashemi, S. Ketabchi, Numerical comparison of smoothing functions for optimal correction of
an infeasible system of absolute value equations, Numer. Algebra Control Optim. 10 (2019) 13–21.
[14] M. Hladik, H. Moosaei, Some notes on the solvability conditions for absolute value equations,
Optim. Lett. 17 (2023) 211–218.
[15] S. L. Hu, Z. H. Hu, A note on absolute value equations, Optim. Lett. 4 (2010) 417–424.

[16] J. Iqbal, A. Iqbal, M. Arif, LevenbergMarquardt method for solving systems of absolute value
equations, J. Comput. Appl. Math. 282 (2015), 134–138.
[17] X. Jiang, Y. Zhang, A smoothing-type algorithm for absolute value equations, J. Ind. Manag. Optim.
9 (2013) 789–798.
[18] Y. F. Ke, C. F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math.
Comput. 311 (2017) 195–202.
[19] S. Ketabchi, H. Moosaei, Minimum norm solution to the absolute value equation in the convex case,
J. Optim. Theory Appl. 154 (2012) 1080–1087.
[20] Y. Li, S. Du, Modified HS conjugate gradient method for solving generalized absolute value equa-
tions, J. Inequal. Appl. 2019 (2019) 68.

[21] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl. 36 (2007) 43–53.
[22] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett. 3
(2009) 101–108.
[23] O. L. Mangasarian, A hybrid algorithm for solving the absolute value equation, Optim. Lett. 9
(2015) 1469–1474.
[24] O. L. Mangasarian, R.R. Meyer, Absolute value equations, Linear Algebra Appl. 419 (2006) 359–
367.
[25] A. Mansoori, M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput.
Appl. Math. 333 (2018) 28–35.
[26] X. H. Miao, J. Yang, S. Hu, A generalized Newton method for absolute value equations associated
with circular cones Appl. Math. Comput. 269 (2015) 155–168.
[27] X. H. Miao, K. Yao, C. Y. Yang, J. S. Chen, Levenberg-Marquardt method for absolute value
equation associated with second-order cone, Numer. Algebra Control Optim. 12 (2022) 47–61.
[28] C. T. Nguyen, B. Saheya, Y. L. Chang, J. S. Chen, Unified smoothing functions for absolute value
equation associated with second-order cone, Appl. Numer. Math. 135 (2019) 206–227.
[29] O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl.
44 (2009) 363–372.
[30] L. Qi, D. Sun, Smoothing functions and smoothing Newton method for complementarity and vari-
ational inequality problems, J. Optim. Theory Appl. 113 (2002) 121–147.
[31] J. Qiu, J. Yu, S. Lian, Smoothing approximation to the new exact penalty function with two param-
eters, Asia-Pac. J. Oper. Res. 38 (2021) 2140010.
[32] J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b, Linear Multilinear Algebra,
52 (2004) 421–426.
[33] B. Saheya, C. T. Nguyen, J. S. Chen, Neural network based on systematically generated smoothing
functions for absolute value equations, J. Appl. Math. Comput. 61 (2019) 533–558.
[34] B. Saheya, C. H. Yu, J. S. Chen, Numerical comparisons based on four smoothing functions for
absolute value equations, J. Appl. Math. Comput. 56 (2018) 131–149.
[35] A. Sahiner, N. Yilmaz, S. A. Ibrahem, Smoothing approximations to non-smooth functions, Journal
of Multidisciplinary Modeling and Optimizaiton 1 (2018) 69–74.
[36] J. Tang, J. Zhou, A quadratically convergent descent method for absolute value equation Ax +
B|x| = b, Oper. Res. Lett., 47 (2019) 229–234.
[37] A. Wang, Y. Cao, J. X. Chen, Modified Newton-type iteration methods for generalized absolute
value equations, J. Optim. Theory Appl. 181 (2019) 216–230.

[38] F. Wang, Z. Yu, C. Gao, A smoothing neural network algorithm for absolute value equations,
Engineering 7 (2015) 567–576.
[39] S.L. Wu, The unique solution of a class of the new generalized absolute value equation, Appl.
Math. Lett. 116 (2021) 107029.
[40] A.E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog. 43 (2010) 731–737.
[41] N. Yilmaz, A. Kayacan, Smoothing LevenbergMarquardt algorithm for solving non-Lipschitz ab-
solute value equations, J. Appl. Anal. 29 (2023) 277–286.
[42] N. Yilmaz, A. Sahiner, New smoothing approximations to piecewise smooth functions and applica-
tions, Numer. Funct. Anal. Optim. 40 (2019) 523–534.
[43] N. Yilmaz, A. Sahiner, Smoothing techniques in solving non-Lipschitz absolute value equations,
Int. J. Comput. Math. 100 (2023) 867–879.
[44] I. Zang, A smoothing out technique for min-max optimization. Math. Program. 19 (1980) 61–77.
[45] L. Zhang, X. Zhang, Global linear and quadratic one-step smoothing Newton method for P0-LCP,
J. Global Optim. 25 (2003) 363–376.