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doi: 10.3934/jimo.2022055
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An inertial inverse-free dynamical system for solving absolute value equations

1. 

Institute for Optimization and Decision Analytics, Liaoning Technical University, Fuxin, 123000, China

2. 

School of Mathematics and Statistics, FJKLMAA and Center for Applied Mathematics of Fujian Province, Fujian Normal University, Fuzhou, 350007, China

3. 

School of Mathematics, Liaoning University, Shenyang, 110036, China

4. 

LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

*Corresponding author: Cairong Chen

Received  November 2021 Revised  March 2022 Early access April 2022

A novel inertial inverse-free dynamical system is proposed to solve the absolute value equation (AVE). Under mild conditions, the proposed model has a unique solution and its solution trajectories asymptotically converge to the solution of the AVE. Compared with the existing dynamical systems, the presence of the inertial term allows the new model to be more convenient for exploring different solutions of the AVE. Numerical results illustrate the effectiveness of the presented method.

Citation: Dongmei Yu, Cairong Chen, Yinong Yang, Deren Han. An inertial inverse-free dynamical system for solving absolute value equations. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022055
References:
[1]

L. AbdallahM. Haddou and T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196-207.  doi: 10.1016/j.cam.2017.06.019.

[2]

L. CaccettaB. Qu and G.-L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45-58.  doi: 10.1007/s10589-009-9242-9.

[3]

C.-R. ChenY.-N. YangD.-M. Yu and D.-R. Han, An inverse-free dynamical system for solving the absolute value equations, Appl. Numer. Math., 168 (2021), 170-181.  doi: 10.1016/j.apnum.2021.06.002.

[4]

C.-R. Chen, D.-M. Yu, D.-R. Han, Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations, preprint, 2020, arXiv: 2001.05781. doi: 10.1016/j.aml.2019.03.033.

[5]

J.Y.B. CruzO.P. Ferreira and L.F. Prudente, On the global convergence of the inexact semi-smooth Newton method for absolute value equation, Comput. Optim. Appl., 65 (2016), 93-108.  doi: 10.1007/s10589-016-9837-x.

[6]

V. EdalatpourD. Hezari and D.K. Salkuyeh, A generalization of the Gauss-Seidel iteration method for solving absolute value equations, Appl. Math. Comput., 293 (2017), 156-167.  doi: 10.1016/j.amc.2016.08.020.

[7]

X.-B. Gao and J. Wang, Analysis and application of a one-layer neural network for solving horizontal linear complementarity problems, Int. J. Conput. Int. Sys., 7 (2014), 724-732. 

[8]

X.-M. GuT.-Z. HuangH.-B. LiS.-F. Wang and L. Li, Two CSCS-based iteration methods for solving absolute value equations, J. Appl. Anal. Comput., 7 (2017), 1336-1356.  doi: 10.11948/2017082.

[9]

P. GuoS.-L Wu and C.-X Li, On the SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107-113.  doi: 10.1016/j.aml.2019.03.033.

[10]

X. He, J.-J. Huang, C.-J. Li, Neural network with added inertia for linear complementarity problem, 15th International Conference on Control, Automation, Robotics and Vision (ICARCV), (2018), 18–21.

[11]

X. HeT.-W. HuangJ.-Z. YuC.-D. Li and C.-J. Li, An inertial projection neural network for solving variational inequalities, IEEE Trans. Cybern., 47 (2017), 809-814. 

[12]

M. Hladík, Bounds for the solutions of absolute value equations, Comput. Optim. Appl., 69 (2018), 243-266.  doi: 10.1007/s10589-017-9939-0.

[13]

X.-J. Huang and B.-T. Cui, Neural network-based method for solving absolute value equations, ICIC-EL, 11 (2017), 853-861. 

[14]

J. IqbalA. Iqbal and M. Arif, Levenberg-Marquardt method for solving systems of absolute value equations, J. Comput. Appl. Math., 282 (2015), 134-138.  doi: 10.1016/j.cam.2014.11.062.

[15]

X.-X. JuC.-D. LiX. He and G. Feng, An inertial projection neural network for solving inverse variational inequalities, Neurocomputing, 406 (2020), 99-105. 

[16]

Y.-F. Ke, The new iteration algorithm for absolute value equation, Appl. Math. Lett., 99 (2020), 105990.  doi: 10.1016/j.aml.2019.07.021.

[17]

Y.-F. Ke and C.-F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195-202.  doi: 10.1016/j.amc.2017.05.035.

[18]

H.K. Khalil, Nonlinear Systems, Prentice-Hall, Michigan, NJ, 1996.

[19]

K. Lachhwani, Application of neural network models for mathematical programming problems: A state of art review, Arch. Comput. Method. Eng., 27 (2020), 171-182.  doi: 10.1007/s11831-018-09309-5.

[20]

O.L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43-53.  doi: 10.1007/s10589-006-0395-5.

[21]

O.L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3-8.  doi: 10.1007/s11590-006-0005-6.

[22]

O.L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101-108.  doi: 10.1007/s11590-008-0094-5.

[23]

O.L. Mangasarian and R.R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359-367.  doi: 10.1016/j.laa.2006.05.004.

[24]

A. Mansoori and M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 333 (2018), 28-35.  doi: 10.1016/j.cam.2017.09.032.

[25]

A. MansooriM. Eshaghnezhad and S. Effati, An efficient neural network model for solving the absolute value equations, IEEE T. Circuits-II, 65 (2017), 391-395. 

[26]

F. Mezzadri, On the solution of general absolute value equations, Appl. Math. Lett., 107 (2020), 106462.  doi: 10.1016/j.aml.2020.106462.

[27]

M.A. NoorJ. IqbalK.I. Noor and E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett., 6 (2012), 1027-1033.  doi: 10.1007/s11590-011-0332-0.

[28]

O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363-372.  doi: 10.1007/s10589-007-9158-1.

[29]

F. Rahpeymaii, K. Amini, T. Allahviranloo, M.R. Malkhalifeh, A new class of conjugate gradient methods for unconstrained smooth optimization and absolute value equations, Calcolo, 56 (2019), 2. Available from: https://doi.org/10.1007/s10092-018-0298-8. doi: 10.1007/s10092-018-0298-8.

[30]

J. Rohn, A theorem of the alternatives for the equation $Ax+ B|x| = b$, Linear Multilinear Algebra, 52 (2004), 421-426.  doi: 10.1080/0308108042000220686.

[31]

B. Saheya, C.-H. Yu, J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, J. Appl. Math. Comput., 56 (2018), 131-149. doi: 10.1007/s12190-016-1065-0.

[32]

A. WangY. Cao and J.-X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 181 (2019), 216-230.  doi: 10.1007/s10957-018-1439-6.

[33]

D.W. Wheeler and W.C. Schieve, Stability and chaos in an inertial two-neuron system, Phys. D Nonlin. Phenom., 105 (1997), 267-284. 

[34]

S.-L. Wu and C.-X. Li, The unique solution of the absolute value equations, Appl. Math. Lett., 76 (2018), 195-200.  doi: 10.1016/j.aml.2017.08.012.

[35]

D.-M. YuC.-R. Chen and D.-R. Han, A modified fixed point iteration method for solving the system of absolute value equations, Optimization, 71 (2022), 449-461.  doi: 10.1080/02331934.2020.1804568.

[36]

M. Zamani and M. Hladík, A new concave minimization algorithm for the absolute value equation solution, Optim. Lett., 15 (2021), 2241-2254.  doi: 10.1007/s11590-020-01691-z.

show all references

References:
[1]

L. AbdallahM. Haddou and T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196-207.  doi: 10.1016/j.cam.2017.06.019.

[2]

L. CaccettaB. Qu and G.-L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45-58.  doi: 10.1007/s10589-009-9242-9.

[3]

C.-R. ChenY.-N. YangD.-M. Yu and D.-R. Han, An inverse-free dynamical system for solving the absolute value equations, Appl. Numer. Math., 168 (2021), 170-181.  doi: 10.1016/j.apnum.2021.06.002.

[4]

C.-R. Chen, D.-M. Yu, D.-R. Han, Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations, preprint, 2020, arXiv: 2001.05781. doi: 10.1016/j.aml.2019.03.033.

[5]

J.Y.B. CruzO.P. Ferreira and L.F. Prudente, On the global convergence of the inexact semi-smooth Newton method for absolute value equation, Comput. Optim. Appl., 65 (2016), 93-108.  doi: 10.1007/s10589-016-9837-x.

[6]

V. EdalatpourD. Hezari and D.K. Salkuyeh, A generalization of the Gauss-Seidel iteration method for solving absolute value equations, Appl. Math. Comput., 293 (2017), 156-167.  doi: 10.1016/j.amc.2016.08.020.

[7]

X.-B. Gao and J. Wang, Analysis and application of a one-layer neural network for solving horizontal linear complementarity problems, Int. J. Conput. Int. Sys., 7 (2014), 724-732. 

[8]

X.-M. GuT.-Z. HuangH.-B. LiS.-F. Wang and L. Li, Two CSCS-based iteration methods for solving absolute value equations, J. Appl. Anal. Comput., 7 (2017), 1336-1356.  doi: 10.11948/2017082.

[9]

P. GuoS.-L Wu and C.-X Li, On the SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107-113.  doi: 10.1016/j.aml.2019.03.033.

[10]

X. He, J.-J. Huang, C.-J. Li, Neural network with added inertia for linear complementarity problem, 15th International Conference on Control, Automation, Robotics and Vision (ICARCV), (2018), 18–21.

[11]

X. HeT.-W. HuangJ.-Z. YuC.-D. Li and C.-J. Li, An inertial projection neural network for solving variational inequalities, IEEE Trans. Cybern., 47 (2017), 809-814. 

[12]

M. Hladík, Bounds for the solutions of absolute value equations, Comput. Optim. Appl., 69 (2018), 243-266.  doi: 10.1007/s10589-017-9939-0.

[13]

X.-J. Huang and B.-T. Cui, Neural network-based method for solving absolute value equations, ICIC-EL, 11 (2017), 853-861. 

[14]

J. IqbalA. Iqbal and M. Arif, Levenberg-Marquardt method for solving systems of absolute value equations, J. Comput. Appl. Math., 282 (2015), 134-138.  doi: 10.1016/j.cam.2014.11.062.

[15]

X.-X. JuC.-D. LiX. He and G. Feng, An inertial projection neural network for solving inverse variational inequalities, Neurocomputing, 406 (2020), 99-105. 

[16]

Y.-F. Ke, The new iteration algorithm for absolute value equation, Appl. Math. Lett., 99 (2020), 105990.  doi: 10.1016/j.aml.2019.07.021.

[17]

Y.-F. Ke and C.-F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195-202.  doi: 10.1016/j.amc.2017.05.035.

[18]

H.K. Khalil, Nonlinear Systems, Prentice-Hall, Michigan, NJ, 1996.

[19]

K. Lachhwani, Application of neural network models for mathematical programming problems: A state of art review, Arch. Comput. Method. Eng., 27 (2020), 171-182.  doi: 10.1007/s11831-018-09309-5.

[20]

O.L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43-53.  doi: 10.1007/s10589-006-0395-5.

[21]

O.L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3-8.  doi: 10.1007/s11590-006-0005-6.

[22]

O.L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101-108.  doi: 10.1007/s11590-008-0094-5.

[23]

O.L. Mangasarian and R.R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359-367.  doi: 10.1016/j.laa.2006.05.004.

[24]

A. Mansoori and M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 333 (2018), 28-35.  doi: 10.1016/j.cam.2017.09.032.

[25]

A. MansooriM. Eshaghnezhad and S. Effati, An efficient neural network model for solving the absolute value equations, IEEE T. Circuits-II, 65 (2017), 391-395. 

[26]

F. Mezzadri, On the solution of general absolute value equations, Appl. Math. Lett., 107 (2020), 106462.  doi: 10.1016/j.aml.2020.106462.

[27]

M.A. NoorJ. IqbalK.I. Noor and E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett., 6 (2012), 1027-1033.  doi: 10.1007/s11590-011-0332-0.

[28]

O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363-372.  doi: 10.1007/s10589-007-9158-1.

[29]

F. Rahpeymaii, K. Amini, T. Allahviranloo, M.R. Malkhalifeh, A new class of conjugate gradient methods for unconstrained smooth optimization and absolute value equations, Calcolo, 56 (2019), 2. Available from: https://doi.org/10.1007/s10092-018-0298-8. doi: 10.1007/s10092-018-0298-8.

[30]

J. Rohn, A theorem of the alternatives for the equation $Ax+ B|x| = b$, Linear Multilinear Algebra, 52 (2004), 421-426.  doi: 10.1080/0308108042000220686.

[31]

B. Saheya, C.-H. Yu, J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, J. Appl. Math. Comput., 56 (2018), 131-149. doi: 10.1007/s12190-016-1065-0.

[32]

A. WangY. Cao and J.-X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 181 (2019), 216-230.  doi: 10.1007/s10957-018-1439-6.

[33]

D.W. Wheeler and W.C. Schieve, Stability and chaos in an inertial two-neuron system, Phys. D Nonlin. Phenom., 105 (1997), 267-284. 

[34]

S.-L. Wu and C.-X. Li, The unique solution of the absolute value equations, Appl. Math. Lett., 76 (2018), 195-200.  doi: 10.1016/j.aml.2017.08.012.

[35]

D.-M. YuC.-R. Chen and D.-R. Han, A modified fixed point iteration method for solving the system of absolute value equations, Optimization, 71 (2022), 449-461.  doi: 10.1080/02331934.2020.1804568.

[36]

M. Zamani and M. Hladík, A new concave minimization algorithm for the absolute value equation solution, Optim. Lett., 15 (2021), 2241-2254.  doi: 10.1007/s11590-020-01691-z.

Figure 1.  Phase diagrams of (4) with different values of $ \lambda $ in Example 2
Table 1.  Numerical results for Example 1
$ x_0 $ method IFDS IIFDS
$ [1, 1]^\top $ Convergence points $ [1.0000, -3.085\times 10^{-10}]^\top $ $ \lambda_0:[1.2, -2.908\times 10^{-11}]^\top $
$ \lambda_1:[1.222, -3.36\times 10^{-11}]^\top $
$ \lambda_2:[1.25, -3.202\times 10^{-11}]^\top $
$ \lambda_3:[1.286, -6.831\times 10^{-11}]^\top $
$ \lambda_4:[1.333, -7.238\times 10^{-11}]^\top $
$ \lambda_5:[1.4, -0.004251]^\top $
$ \lambda_6:[1.5, -0.07681]^\top $
$ \lambda_7:[1.667, -0.2886]^\top $
$ \lambda_8:[2, -0.8016]^\top $
$ \lambda_9:[3, -2.474]^\top $
$ [1, -1]^\top $ Convergence points $ [1, -1]^\top $ $ \lambda_0:[1.2, -1.2]^\top $
$ \lambda_1:[1.222, -1.222]^\top $
$ \lambda_2:[1.25, -1.25]^\top $
$ \lambda_3:[1.286, -1.286]^\top $
$ \lambda_4:[1.333, -1.333]^\top $
$ \lambda_5:[1.4, -1.4]^\top $
$ \lambda_6:[1.5, -1.5]^\top $
$ \lambda_7:[1.667, -1.667]^\top $
$ \lambda_8:[2, -2]^\top $
$ \lambda_9:[3, -3]^\top $
$ [-1, 1]^\top $ Convergence points $ [3.085\times 10^{-10}, -3.085\times 10^{-10}]^\top $ $ \lambda_0:[2.908\times 10^{-11}, -2.908\times 10^{-11}]^\top $
$ \lambda_1:[3.36\times 10^{-11}, -3.36\times 10^{-11}]^\top $
$ \lambda_2:[3.202\times 10^{-11}, -3.202\times 10^{-11}]^\top $
$ \lambda_3:[6.831\times 10^{-11}, -6.831\times 10^{-11}]^\top $
$ \lambda_4:[7.238\times 10^{-11}, -7.238\times 10^{-11}]^\top $
$ \lambda_5:[0.004251, -0.004251]^\top $
$ \lambda_6:[0.07681, -07681]^\top $
$ \lambda_7:[0.2886, -0.2886]^\top $
$ \lambda_8:[0.8061, -0.8061]^\top $
$ \lambda_9:[2.474, -2.474]^\top $
$ [-1, -1]^\top $ Convergence points $ [3.085\times 10^{-10}, -1]^\top $ $ \lambda_0:[2.908\times 10^{-11}, -1.2]^\top $
$ \lambda_1:[3.36\times 10^{-11}, -1.222]^\top $
$ \lambda_2:[3.202\times 10^{-11}, -1.25]^\top $
$ \lambda_3:[6.831\times 10^{-11}, -1.286]^\top $
$ \lambda_4:[7.238\times 10^{-11}, -1.333]^\top $
$ \lambda_5:[0.004251, -1.4]^\top $
$ \lambda_6:[0.07681, -1.5]^\top $
$ \lambda_7:[0.2886, -1.667]^\top $
$ \lambda_8:[0.8016, -2]^\top $
$ \lambda_9:[2.474, -3]^\top $
$ x_0 $ method IFDS IIFDS
$ [1, 1]^\top $ Convergence points $ [1.0000, -3.085\times 10^{-10}]^\top $ $ \lambda_0:[1.2, -2.908\times 10^{-11}]^\top $
$ \lambda_1:[1.222, -3.36\times 10^{-11}]^\top $
$ \lambda_2:[1.25, -3.202\times 10^{-11}]^\top $
$ \lambda_3:[1.286, -6.831\times 10^{-11}]^\top $
$ \lambda_4:[1.333, -7.238\times 10^{-11}]^\top $
$ \lambda_5:[1.4, -0.004251]^\top $
$ \lambda_6:[1.5, -0.07681]^\top $
$ \lambda_7:[1.667, -0.2886]^\top $
$ \lambda_8:[2, -0.8016]^\top $
$ \lambda_9:[3, -2.474]^\top $
$ [1, -1]^\top $ Convergence points $ [1, -1]^\top $ $ \lambda_0:[1.2, -1.2]^\top $
$ \lambda_1:[1.222, -1.222]^\top $
$ \lambda_2:[1.25, -1.25]^\top $
$ \lambda_3:[1.286, -1.286]^\top $
$ \lambda_4:[1.333, -1.333]^\top $
$ \lambda_5:[1.4, -1.4]^\top $
$ \lambda_6:[1.5, -1.5]^\top $
$ \lambda_7:[1.667, -1.667]^\top $
$ \lambda_8:[2, -2]^\top $
$ \lambda_9:[3, -3]^\top $
$ [-1, 1]^\top $ Convergence points $ [3.085\times 10^{-10}, -3.085\times 10^{-10}]^\top $ $ \lambda_0:[2.908\times 10^{-11}, -2.908\times 10^{-11}]^\top $
$ \lambda_1:[3.36\times 10^{-11}, -3.36\times 10^{-11}]^\top $
$ \lambda_2:[3.202\times 10^{-11}, -3.202\times 10^{-11}]^\top $
$ \lambda_3:[6.831\times 10^{-11}, -6.831\times 10^{-11}]^\top $
$ \lambda_4:[7.238\times 10^{-11}, -7.238\times 10^{-11}]^\top $
$ \lambda_5:[0.004251, -0.004251]^\top $
$ \lambda_6:[0.07681, -07681]^\top $
$ \lambda_7:[0.2886, -0.2886]^\top $
$ \lambda_8:[0.8061, -0.8061]^\top $
$ \lambda_9:[2.474, -2.474]^\top $
$ [-1, -1]^\top $ Convergence points $ [3.085\times 10^{-10}, -1]^\top $ $ \lambda_0:[2.908\times 10^{-11}, -1.2]^\top $
$ \lambda_1:[3.36\times 10^{-11}, -1.222]^\top $
$ \lambda_2:[3.202\times 10^{-11}, -1.25]^\top $
$ \lambda_3:[6.831\times 10^{-11}, -1.286]^\top $
$ \lambda_4:[7.238\times 10^{-11}, -1.333]^\top $
$ \lambda_5:[0.004251, -1.4]^\top $
$ \lambda_6:[0.07681, -1.5]^\top $
$ \lambda_7:[0.2886, -1.667]^\top $
$ \lambda_8:[0.8016, -2]^\top $
$ \lambda_9:[2.474, -3]^\top $
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