doi: 10.3934/jimo.2021150
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A new smoothing spectral conjugate gradient method for solving tensor complementarity problems

School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

* Corresponding author: Shou-Qiang Du

Received  December 2020 Revised  May 2021 Early access September 2021

In recent years, the tensor complementarity problem has attracted widespread attention and has been extensively studied. The research work of tensor complementarity problem mainly focused on theory, solution methods and applications. In this paper, we study the solution method of tensor complementarity problem. Based on the equivalence relation of the tensor complementarity problem and unconstrained optimization problem, we propose a new smoothing spectral conjugate gradient method with Armijo line search. Under mild conditions, we establish the global convergence of the proposed method. Finally, some numerical results are given to show the effectiveness of the proposed method and verify our theoretical results.

Citation: ShiChun Lv, Shou-Qiang Du. A new smoothing spectral conjugate gradient method for solving tensor complementarity problems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021150
References:
[1]

M. Al-BaaliY. Narushima and H. Yabe, A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Comput. Optim. Appl., 60 (2015), 89-110.  doi: 10.1007/s10589-014-9662-z.  Google Scholar

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M. AliyuP. Kumam and B. Auwal, A modified conjugate gradient method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 145 (2019), 507-520.  doi: 10.1016/j.apnum.2019.05.012.  Google Scholar

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S. Bojari and M. Eslahchi, Two families of scaled three-term conjugate gradient methods with sufficient descent property for nonconvex optimization, Numer. Algor., 83 (2020), 901-933.  doi: 10.1007/s11075-019-00709-7.  Google Scholar

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C. Chen and L. Zhang, Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem, Appl. Number. Math., 145 (2019), 458-468.  doi: 10.1016/j.apnum.2019.05.013.  Google Scholar

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[31]

C. LingH. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Comput. Optim. Appl., 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z.  Google Scholar

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[33]

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D. LiuW. Li and S. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear Multilinear A., 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.  Google Scholar

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J. LiuS. Du and Y. Chen, A sufficient descent nonlinear conjugate gradient method for solving $\mathcal M$-tensor equations, J. Comput. Appl., 371 (2020), 112709.  doi: 10.1016/j.cam.2019.112709.  Google Scholar

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show all references

References:
[1]

M. Al-BaaliY. Narushima and H. Yabe, A family of three-term conjugate gradient methods with sufficient descent property for unconstrained optimization, Comput. Optim. Appl., 60 (2015), 89-110.  doi: 10.1007/s10589-014-9662-z.  Google Scholar

[2]

M. AliyuP. Kumam and B. Auwal, A modified conjugate gradient method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 145 (2019), 507-520.  doi: 10.1016/j.apnum.2019.05.012.  Google Scholar

[3]

S. Babaie-KafakiR. Ghanbari and N. Mahdavi-Amiri, Two new conjugate gradient methods based on modified secant equations, J. Comput. Appl. Math., 234 (2010), 1374-1386.  doi: 10.1016/j.cam.2010.01.052.  Google Scholar

[4]

X. BaiZ. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, J. Optim. Theory Appl., 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.  Google Scholar

[5]

J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[6]

E. Birgin and J. Martínez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim., 43 (2001), 117-128.  doi: 10.1007/s00245-001-0003-0.  Google Scholar

[7]

S. Bojari and M. Eslahchi, Two families of scaled three-term conjugate gradient methods with sufficient descent property for nonconvex optimization, Numer. Algor., 83 (2020), 901-933.  doi: 10.1007/s11075-019-00709-7.  Google Scholar

[8]

K. ChangK. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.  doi: 10.4310/CMS.2008.v6.n2.a12.  Google Scholar

[9]

M. CheL. Qi and Y. Wei, Positive-definite tensors to nonlinear complementarity problems, J. Optim. Theory Appl., 168 (2016), 475-487.  doi: 10.1007/s10957-015-0773-1.  Google Scholar

[10]

B. Chen and P. Harker, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), 403-420.  doi: 10.1137/S1052623495280615.  Google Scholar

[11]

C. Chen and L. Zhang, Finding Nash equilibrium for a class of multi-person noncooperative games via solving tensor complementarity problem, Appl. Number. Math., 145 (2019), 458-468.  doi: 10.1016/j.apnum.2019.05.013.  Google Scholar

[12]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[13]

Y. DaiL. Liao and D. Li, On restart procedures for the conjugate gradient method, Numer. Algor., 35 (2004), 249-260.  doi: 10.1023/B:NUMA.0000021761.10993.6e.  Google Scholar

[14]

W. DingL. Qi and Y. Wei, $\mathcal M$-tensors and nonsingular $\mathcal M$-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.  Google Scholar

[15]

W. Ding and Y. Wei, Solving multi-linear systems with $\mathcal M$-tensors, J. Sci. Comput., 68 (2016), 689-715.  doi: 10.1007/s10915-015-0156-7.  Google Scholar

[16]

W. DingZ. Luo and L. Qi, $\mathcal P$-tensors, $\mathcal P_0$-tensors, and their applications, Linear Algebra Appl., 555 (2018), 336-354.  doi: 10.1016/j.laa.2018.06.028.  Google Scholar

[17]

S. DuL. ZhangC. Chen and L. Qi, Tensor absolute value equations, Sci. China Math., 61 (2018), 1695-1710.  doi: 10.1007/s11425-017-9238-6.  Google Scholar

[18]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Glob. Optim., 73 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.  Google Scholar

[19]

S. DuM. Che and Y. Wei, Stochastic structured tensors to stochastic complementarity problems, Comput. Optim. Appl., 75 (2020), 649-668.  doi: 10.1007/s10589-019-00144-3.  Google Scholar

[20]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149-154.  doi: 10.1093/comjnl/7.2.149.  Google Scholar

[21]

M. Gowda, Z. Luo, L. Qi and N. Xiu, $\mathcal Z$-tensors and complementarity problems, arXiv: 1510.07933v2 Google Scholar

[22]

W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.  Google Scholar

[23]

L. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949-963.  doi: 10.1007/s10957-018-1422-2.  Google Scholar

[24]

M. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436.  doi: 10.6028/jres.049.044.  Google Scholar

[25]

S. HuZ. HuangC. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symb. Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar

[26]

Z. Huang and L. Qi, Formulating an n-person noncoorperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.  Google Scholar

[27]

Z. Huang and L. Qi, Tensor complementarity problems–Part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.  Google Scholar

[28]

Z. Huang and L. Qi, Tensor complementarity problems–Part III: Applications, J. Optim. Theory Appl., 183 (2019), 771-791.  doi: 10.1007/s10957-019-01573-0.  Google Scholar

[29]

X. Jiang and J. Jian, Improved Fletcher-Reeves and Dai-Yuan conjugate gradient methods with the strong Wolfe line search, J. Comput. Appl. Math., 348 (2019), 525-534.  doi: 10.1016/j.cam.2018.09.012.  Google Scholar

[30]

Y. LiS. Du and L. Zhang, Tensor quadratic eigenvalue complementarity problem, Pac. J. Optim., 17 (2021), 251-268.   Google Scholar

[31]

C. LingH. He and L. Qi, On the cone eigenvalue complementarity problem for higher-order tensors, Comput. Optim. Appl., 63 (2016), 143-168.  doi: 10.1007/s10589-015-9767-z.  Google Scholar

[32]

C. LingH. He and L. Qi, Higher-degree eigenvalue complementarity problems for tensors, Comput. Optim. Appl., 64 (2016), 149-176.  doi: 10.1007/s10589-015-9805-x.  Google Scholar

[33]

C. LingW. YanH. He and L. Qi, Further study on tensor absolute value equations, Sci. China Math., 63 (2020), 2137-2156.  doi: 10.1007/s11425-018-9560-3.  Google Scholar

[34]

D. LiuW. Li and S. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear Multilinear A., 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.  Google Scholar

[35]

J. LiuS. Du and Y. Chen, A sufficient descent nonlinear conjugate gradient method for solving $\mathcal M$-tensor equations, J. Comput. Appl., 371 (2020), 112709.  doi: 10.1016/j.cam.2019.112709.  Google Scholar

[36]

I. Livieris and P. Pintelas, A new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization, J. Comput. Appl. Math., 239 (2013), 396-405.  doi: 10.1016/j.cam.2012.09.007.  Google Scholar

[37]

Z. LuoL. Qi and N. Xiu, The sparsest solutions to $\mathcal Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.  Google Scholar

[38]

G. Meurant, On prescribing the convergence behavior of the conjugate gradient algorithm, Numer. Algor., 84 (2020), 1353-1380.  doi: 10.1007/s11075-019-00851-2.  Google Scholar

[39]

Q. Ni and L. Qi, A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map, J. Global Optim., 61 (2015), 627-641.  doi: 10.1007/s10898-014-0209-8.  Google Scholar

[40]

M. Powell, Restart procedures for the conjugate gradient method, Math. Program., 12 (1977), 241-254.  doi: 10.1007/BF01593790.  Google Scholar

[41]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[42]

L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.  doi: 10.1016/j.laa.2013.03.015.  Google Scholar

[43]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[44]

L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Advances in Mechanics and Mathematics, 39. Springer, Singapore, 2018. doi: 10.1007/978-981-10-8058-6.  Google Scholar

[45]

L. Qi and Z. Huang, Tensor complementarity problems–Part II: Solution methods, J. Optim. Theory Appl., 183 (2019), 365-385.  doi: 10.1007/s10957-019-01568-x.  Google Scholar

[46]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.  doi: 10.1007/s10957-014-0616-5.  Google Scholar

[47]

Y. Song and L. Qi, Tensor complementarity problem and semi-positive tensors, J. Optim. Theory Appl., 169 (2016), 1069-1078.  doi: 10.1007/s10957-015-0800-2.  Google Scholar

[48]

Y. Song and L. Qi, Properties of tensor complementarity problem and some classes of structured tensors, Ann. Appl. Math., 33 (2017), 308-323.   Google Scholar

[49]

Z. WanZ. Yang and Y. Wang, New spectral PRP conjugate gradient method for unconstrained optimization, Appl. Math. Lett., 24 (2011), 16-22.  doi: 10.1016/j.aml.2010.08.002.  Google Scholar

[50]

X. WangM. Che and Y. Wei, Global uniqueness and solvability of tensor complementarity problems for $\mathcal{H}_+$-tensors, Numer. Algorithms, 84 (2020), 567-590.  doi: 10.1007/s11075-019-00769-9.  Google Scholar

[51]

X. Wang, M. Che and Y. Wei, Preconditioned tensor splitting AOR iterative methods for $\mathcal H$-tensor equations,, Numer. Linear Algebra Appl., 27 (2020), e2329. doi: 10.1002/nla.2329.  Google Scholar

[52] Y. Wei and W. Ding, Theory and Computation of Tensors: Multi-Dimensional Arrays, Academic Press, London, 2016.   Google Scholar
[53]

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Figure 1.  Numerical results of Example 4.1 with different vector $ q $
Figure 2.  Numerical results of Example 4.2 with different vector $ q $
Figure 3.  Numerical results of Example 4.3 with different vector $ q $
Figure 4.  Numerical results of Example 4.4 with different initial points
Figure 5.  Numerical results of Example 4.5 with different initial points
Table 1.  The numerical results of Example 4.1
x0 Sol Val
$ (0.8333, 0.2037, 0.5444, 0.8749)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 1.9158e-15
$ (0.9460, 0.0916, 0.9084, 0.5100)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.9449e-14
$ (0.1445, 0.3705, 0.6224, 0.9976)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.1031e-15
$ (0.6966, 0.0646, 0.7477, 0.4204)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 8.9036e-14
$ (0.8113, 0.3796, 0.3191, 0.9861)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 4.1630e-14
x0 Sol Val
$ (0.8333, 0.2037, 0.5444, 0.8749)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 1.9158e-15
$ (0.9460, 0.0916, 0.9084, 0.5100)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.9449e-14
$ (0.1445, 0.3705, 0.6224, 0.9976)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 2.1031e-15
$ (0.6966, 0.0646, 0.7477, 0.4204)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 8.9036e-14
$ (0.8113, 0.3796, 0.3191, 0.9861)^T $ $ (0.0000, 0.0000, 0.0000, 0.0000)^T $ 4.1630e-14
Table 2.  The numerical results of Example 4.2
q Sol K Val
$ (1, 2, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 7 2.8222e-13
$ (1, -2, 3)^T $ $ (0.0000, 1.0000, 0.0000)^T $ 70 3.1532e-13
$ (3, 3, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 6 2.1459e-14
$ (-3, -2, -3)^T $ $ (1.3161, 1.0000, 1.0000)^T $ 16 1.0256e-15
$ (-3, -1, -2)^T $ $ (1.3161, 0.8409, 0.9036)^T $ 19 1.1984e-14
q Sol K Val
$ (1, 2, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 7 2.8222e-13
$ (1, -2, 3)^T $ $ (0.0000, 1.0000, 0.0000)^T $ 70 3.1532e-13
$ (3, 3, 3)^T $ $ (0.0000, 0.0000, 0.0000)^T $ 6 2.1459e-14
$ (-3, -2, -3)^T $ $ (1.3161, 1.0000, 1.0000)^T $ 16 1.0256e-15
$ (-3, -1, -2)^T $ $ (1.3161, 0.8409, 0.9036)^T $ 19 1.1984e-14
Table 3.  The numerical results of Example 4.3
q Sol K Val
$ (5, 3)^T $ $ (0.0000, 0.0000)^T $ 7 8.9458e-14
$ (2, -3)^T $ $ (0.0000, 1.7321)^T $ 76 2.6588e-13
$ (-5, -3)^T $ $ (0.3127, 1.9233)^T $ 30 9.4915e-14
$ (-5, 3)^T $ $ (1.5513, 0.6847)^T $ 55 2.0370e-14
$ (0, -5)^T $ $ (0.0000, 2.2361)^T $ 34 7.5750e-14
q Sol K Val
$ (5, 3)^T $ $ (0.0000, 0.0000)^T $ 7 8.9458e-14
$ (2, -3)^T $ $ (0.0000, 1.7321)^T $ 76 2.6588e-13
$ (-5, -3)^T $ $ (0.3127, 1.9233)^T $ 30 9.4915e-14
$ (-5, 3)^T $ $ (1.5513, 0.6847)^T $ 55 2.0370e-14
$ (0, -5)^T $ $ (0.0000, 2.2361)^T $ 34 7.5750e-14
Table 4.  The numerical results of Example 4.4
Alg. x0 Sol K Val
Algorithm 3.1 $ (0.3804, 0.5678)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
Algorithm 3.1 $ (0.0759, 0.0540)^T $ $ (0.0000, 0.0000)^T $ 7 2.9600e-13
Algorithm 3.1 $ (0.9340, 0.1299)^T $ $ (0.0000, 0.0000)^T $ 6 9.5895e-15
Algorithm 3.1 $ (0.1622, 0.7943)^T $ $ (0.0000, 0.0000)^T $ 6 4.5196e-13
Algorithm 3.1 $ (0.3112, 0.5285)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
MIP - $ (0.0000, 0.0000)^T $ 39 -
MIP - $ (0.0000, 0.0000)^T $ 29 -
Alg. x0 Sol K Val
Algorithm 3.1 $ (0.3804, 0.5678)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
Algorithm 3.1 $ (0.0759, 0.0540)^T $ $ (0.0000, 0.0000)^T $ 7 2.9600e-13
Algorithm 3.1 $ (0.9340, 0.1299)^T $ $ (0.0000, 0.0000)^T $ 6 9.5895e-15
Algorithm 3.1 $ (0.1622, 0.7943)^T $ $ (0.0000, 0.0000)^T $ 6 4.5196e-13
Algorithm 3.1 $ (0.3112, 0.5285)^T $ $ (0.0000, 0.0000)^T $ 7 2.9601e-13
MIP - $ (0.0000, 0.0000)^T $ 39 -
MIP - $ (0.0000, 0.0000)^T $ 29 -
Table 5.  The numerical results of Example 4.5
x0 K Val
$ (0.7655, 0.7951, 0.1868, 0.4897, 0.4455, 0.6463)^T $ 10 3.0376e-13
$ (0.7094, 0.7547, 0.2760, 0.6797, 0.6551, 0.1626)^T $ 12 2.7860e-15
$ (0.5472, 0.1386, 0.1493, 0.2575, 0.8407, 0.2543)^T $ 10 1.9144e-13
$ (0.8143, 0.2435, 0.9293, 0.3500, 0.1966, 0.2511)^T $ 10 1.7226e-13
$ (0.6160, 0.4733, 0.3517, 0.8308, 0.5853, 0.5497)^T $ 9 1.2683e-14
x0 K Val
$ (0.7655, 0.7951, 0.1868, 0.4897, 0.4455, 0.6463)^T $ 10 3.0376e-13
$ (0.7094, 0.7547, 0.2760, 0.6797, 0.6551, 0.1626)^T $ 12 2.7860e-15
$ (0.5472, 0.1386, 0.1493, 0.2575, 0.8407, 0.2543)^T $ 10 1.9144e-13
$ (0.8143, 0.2435, 0.9293, 0.3500, 0.1966, 0.2511)^T $ 10 1.7226e-13
$ (0.6160, 0.4733, 0.3517, 0.8308, 0.5853, 0.5497)^T $ 9 1.2683e-14
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