Top

previous article A new modified three-step iteration method for ...

next article Numerical solution of Itô-Volterra integral equ...

Hint

Swipe to navigate through the articles of this issue

16-07-2019 | Original Paper | Issue 2/2020

Global uniqueness and solvability of tensor complementarity problems for \(\mathcal {H}_{+}\)-tensors

Journal:: Numerical Algorithms > Issue 2/2020

Authors:: Xuezhong Wang, Maolin Che, Yimin Wei

» Get access to the full-text

Important notes

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Abstract

Inthis paper, we study the global uniqueness and solvability of tensor complementarity problems for \(\mathcal {H}_{+}\)-tensors. We obtain a sufficient condition of the global uniqueness and solvability of tensor complementarity problems for \(\mathcal {H}_{+}\)-tensors. We present nonlinear dynamical system models for solving the tensor complementarity problem (TCP). We prove that the presented dynamical system models are stable in the sense of Lyapunov stability theory for considering three classes of structured tensors. The computer simulation results further substantiate that the considered dynamical system can be used to solve the TCP.

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 58.000 Bücher
über 300 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 50.000 Bücher
über 380 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Maschinenbau + Werkstoffe

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 69.000 Bücher
über 500 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Bai, X., Huang, Z., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016) MathSciNetMATH

Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994) MATH

Bonnans, J., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–492 (1999) MathSciNetMATH

Bouzerdoum, A., Pattison, T.: Neural network for quadratic optimization with bound constraints. IEEE Trans. Neural Netw. 4, 293–304 (1993)

Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475–487 (2016) MathSciNetMATH

Che, M., Qi, L., Wei, Y.: Stochastic r ₀ tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019) MathSciNetMATH

Chen, H., Li, G., Qi, L.: Sum-of-squares tensors and their sum-of-squares rank. arXiv: 1504.03414V1 (2015)

Chua, L., Lin, G.: Nonlinear programming without computation. IEEE Transactions on Circuits and Systems 31, 182–188 (1984) MathSciNet

Clarke, F.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990) MATH

10.

Cottle, R.: Linear Complementarity Problem. Academic Press Inc., New York (1992) MATH

11.

De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996) MathSciNetMATH

12.

Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45, 31–43 (2006) MathSciNetMATH

13.

Ding, W., Luo, Z., Qi, L.: P-tensors, P ₀-tensors, and tensor complementarity problem. Linear Algebra Appl. 555, 336–354 (2018) MathSciNet

14.

Ding, W., Qi, L., Wei, Y.: \(\mathcal {M}\)-tensors and nonsingular \(\mathcal {M}\)-tensors. Linear Algebra Appl. 439, 3264–3278 (2013) MathSciNetMATH

15.

Ding, W., Wei, Y.: Solving multi-linear systems with \(\mathcal {M}\)-tensors. J. Sci. Comput. 68, 689–715 (2016) MathSciNetMATH

16.

Du, S., Zhang, L.: A mixed integer programming approach to the tensor complementarity problem. J. Glob. Optim. 73, 789–800 (2019) MathSciNetMATH

17.

Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci China Math 61, 1695–1710 (2018) MathSciNetMATH

18.

Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Singapore, New York (2003) MATH

19.

Fischer, A.: A special newton-type optimization method. Optimization 24, 269–284 (1992) MathSciNetMATH

20.

Gowda, M., Luo, Z., Qi, L., Xiu, N.: \(\mathcal {Z}\)-tensors and complementarity problems. arXiv: 1510.07933

21.

Han, L.: A homotopy method for solving multilinear systems with \(\mathcal {M}\)-tensors. Appl. Math. Lett. 69, 49–54 (2017) MathSciNetMATH

22.

Han, L.: A continuation method for tensor complementarity problems. J. Optim. Theory Appl. 180, 949–963 (2019) MathSciNetMATH

23.

Hopfield, J., Tank, D.: Neural computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985) MATH

24.

Huang, Z., Qi, L.: Formulating an n-person noncooperative game as a tensor complementarity problem. Comput. Optim. Appl. 66, 557–576 (2017) MathSciNetMATH

25.

Kannan, M., Shaked-Monderer, N., Berman, A.: Some properties of strong \(\mathcal {H}\)-tensors and general \(\mathcal {H}\)-tensors. Linear Algebra Appl. 476, 42–55 (2015) MathSciNetMATH

26.

Li, S., Chen, S., Liu, B.: Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function. Neural. Process. Lett. 37, 189–205 (2013)

27.

Liao, L., Qi, H.: A neural network for the linear complementarity problem. Math. Comput. Model. 29, 9–18 (1999) MathSciNetMATH

28.

Liao, L., Qi, H., Qi, L.: Solving nonlinear complementarity problems with neural networks: a reformulation method approach. J. Comput. Appl. Math. 131, 343–359 (2001) MathSciNetMATH

29.

Liu, D., Li, W., Vong, S.: Tensor complementarity problems: the GUS-property and an algorithm. Linear and Multilinear Algebra 28, 1726–1749 (2018) MathSciNetMATH

30.

Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \(\mathcal {Z}\)-tensor complementarity problems. Optim. Lett. 11, 471–482 (2017) MathSciNetMATH

31.

More, J.: Global methods for nonlinear complementarity problems. Math. Oper. Res. 21, 589–614 (1996) MathSciNetMATH

32.

Murty, K.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann (1988)

33.

Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005) MathSciNetMATH

34.

Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 33, 353–367 (1993) MathSciNetMATH

35.

Qi, L., Yin, H.: A strongly semismooth integral function and its application. Comput. Optim. Appl. 25, 223–246 (2003) MathSciNetMATH

36.

Qiao, S., Wang, X., Wei, Y.: Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. Linear Algebra Appl. 542, 101–117 (2018) MathSciNetMATH

37.

Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73, 1399–1421 (2013) MathSciNetMATH

38.

Rodriguez-Vazquez, A., Dominguez-Castro, R., Rueda, A., Huertas, J.L., Sanchez-Sinencio, E.: Nonlinear switched capacitor ‘neural’ networks for optimization problems. IEEE Transactions on Circuits and Systems 37, 384–398 (1990) MathSciNet

39.

Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015) MathSciNetMATH

40.

Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1069–1078 (2016) MathSciNetMATH

41.

Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math., pp. 308–323 (2017)

42.

Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170, 85–96 (2016) MathSciNetMATH

43.

Stanimirović, P., Katsikis, V., Li, S.: Hybrid GNN-ZNN models for solving linear matrix equations. Neurocomputing 316, 124–134 (2018)

44.

Stanimirović, P., Petković, M., Gerontitis, D.: Gradient neural network with nonlinear activation for computing inner inverses and the Drazin inverse. Neural. Process. Lett. 48, 109–133 (2018)

45.

Stanimirović, P., Zivković, I., Wei, Y.: Recurrent neural network for computing the Drazin inverse. IEEE Transactions on Neural Networks and Learning Systems 26, 2830–2843 (2015) MathSciNet

46.

Wang, X., Che, M., Qi, L., Wei, Y.: Modified gradient dynamic approach to the tensor complementarity problem. Optim. Methods Softw. https://doi.org/10.1080/10556788.2019.1578766

47.

Wang, X., Che, M., Wei, Y.: Existence and uniqueness of positive solution for \(\mathcal {H}^{+}\)-tensor equations. Appl. Math. Lett. 98, 191–198 (2019) MathSciNet

48.

Wang, X., Che, M., Wei, Y.: Neural networks based approach solving multi-linear systems with \(\mathcal {M}\)-tensors. Neurocomputing 351, 33–42 (2019)

49.

Wang, X., Ma, H., Stanimirović, P.: Nonlinearly activated recurrent neural network for computing the Drazin inverse. Neural. Process. Lett. 46, 195–217 (2017)

50.

Wang, X., Stanimirovic, P., Wei, Y.: Complex ZFs for computing time-varying complex outer inverses. Neurocomputing 275, 983–1001 (2018)

51.

Wang, X., Wei, Y.: Bounds for eigenvalues of nonsingular \(\mathcal {H}\)-tensor. Electronic Journal of Linear Algebra 29, 3–16 (2015) MathSciNetMATH

52.

Wang, X., Wei, Y.: 6Pt width \(\mathcal {H}\)-tensors and nonsingular \(\mathcal {H}\)-tensors. Frontiers of Mathematics in China 11, 557–575 (2016) MathSciNet

53.

Wang, Y., Huang, Z., Bai, X.: Exceptionally regular tensors and tensor complementarity problems. Optim. Methods Softw. 31, 815–828 (2016) MathSciNetMATH

54.

Wang, Y., Zhou, G., Caccetta, L.: Nonsingular \(\mathcal {H}\)-tensor and its criteria. Journal of Industrial and Management Optimization 12, 1173–1186 (2017) MathSciNetMATH

55.

Xie, S., Li, D., Xu, H.: An iterative method for finding the least solution to the tensor complementarity problem. J. Optim. Theory Appl. 175, 119–136 (2017) MathSciNetMATH

56.

Xu, H., Li, D., Xie, S.: An equivalent tensor equation to the tensor complementarity problem with positive semi-definite Z-tensor. Optim. Lett. 13, 685–694 (2019) MathSciNetMATH

57.

Zabczyk, J.: Mathematical Control Theory: an Introduction. Springer, New York (2009) MATH

58.

Zak, S., Upatising, V., Hui, S.: Solving linear programming problems with neural networks: a comparative study. IEEE Trans. Neural Netw. 6, 94–104 (1995)

59.

Zhang, Y.: A set of nonlinear equations and inequalities arising in robotics and its online solution via a primal neural network. Neurocomputing 70, 513–524 (2006)

60.

Zhang, Y., Chen, Z., Chen, K.: Convergence properties analysis of gradient neural network for solving online linear equations. Acta Automat. Sin. 35, 1136–1139 (2009)

61.

Zhang, Y., Shi, Y., Chen, K., Wang, C.: Global exponential convergence and stability of gradient-based neural network for online matrix inversion. Appl. Math. Comput. 215, 1301–1306 (2009) MathSciNetMATH

Title: Global uniqueness and solvability of tensor complementarity problems for -tensors
Authors:: Xuezhong Wang
Maolin Che
Yimin Wei
Publication date: 16-07-2019
DOI: https://doi.org/10.1007/s11075-019-00769-9
Publisher: Springer US
Journal: Numerical Algorithms
Issue 2/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265

Premium Partner

Image Credits

Springer Professional

Hint

Swipe to navigate through the articles of this issue

Publisher’s note

Abstract

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 2/2020

Parallelization, initialization, and boundary treatments for the diamond scheme

Numerical solution of Itô-Volterra integral equation by least squares method

Three-step alternating iterations for index 1 and non-singular matrices

Further comment on another hybrid conjugate gradient algorithm for unconstrained optimization by Andrei

Low-rank updates and divide-and-conquer methods for quadratic matrix equations

The global convergence of the BFGS method under a modified Yuan-Wei-Lu line search technique

Premium Partner