Abstract
Tensors (hypermatrices) are multidimensional analogs of matrices. The tensor complementarity problem is a class of nonlinear complementarity problems with the involved function being defined by a tensor, which is also a direct and natural extension of the linear complementarity problem. In the last few years, the tensor complementarity problem has attracted a lot of attention, and has been studied extensively, from theory to solution methods and applications. This work, with its three parts, aims at contributing to review the state-of-the-art of studies for the tensor complementarity problem and related models. In this part, we describe the theoretical developments for the tensor complementarity problem and related models, including the nonemptiness and compactness of the solution set, global uniqueness and solvability, error bound theory, stability and continuity analysis, and so on. The developments of solution methods and applications for the tensor complementarity problem are given in the second part and the third part, respectively. Some further issues are proposed in all the parts.
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References
Harker, P.T., Pang, J.-S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)
Han, J., Xiu, N., Qi, H.D.: Nonlinear Complementarity Theory and Algorithms. Shanghai Science and Technology Press, Shanghai (2006). (in Chinese)
Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)
Yang, Y., Yang, Q.: A Study on Eigenvalues of Higher-Order Tensors and Related Polynomial Optimization Problems. Science Press, Beijing (2015)
Wei, Y., Ding, W.: Theory and Computation of Tensors: Multi-Dimensional Arrays. Academic Press, London (2016)
Song, Y., Qi, L.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165, 854–873 (2015)
Song, Y., Qi, L.: Properties of tensor complementarity problem and some classes of structured tensors. Ann. Appl. Math. 33, 308–323 (2017)
Che, M., Qi, L., Wei, Y.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168(2), 475–487 (2016)
Qi, L., Chen, H.B., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Berlin (2018)
Song, Y., Yu, G.: Properties of solution set of tensor complementarity problem. J. Optim. Theory Appl. 170(1), 85–96 (2016)
Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
Qi, L.: Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl. 439, 228–238 (2013)
Ding, W., Luo, Z., Qi, L.: \(P\)-tensors, \(P_0\)-tensors, and their applications. Linear Algebra Appl. 555, 336–354 (2018)
Wang, Y., Huang, Z.H., Bai, X.L.: Exceptionally regular tensors and tensor complementarity problems. Optim. Method Softw. 31(4), 815–828 (2016)
Ding, W., Qi, L., Wei, Y.: \(M\)-tensors and nonsingular \(M\)-tensors. Linear Algebra Appl. 439(10), 3264–3278 (2013)
Zhang, L., Qi, L., Zhou, G.: \(M\)-tensors and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437–452 (2014)
Bai, X.L., Huang, Z.H., Wang, Y.: Global uniqueness and solvability for tensor complementarity problems. J. Optim. Theory Appl. 170, 72–84 (2016)
Huang, Z.H., Suo, Y.Y., Wang, J.: On \(Q\)-tensors. arXiv:1509.03088. To appear in Pac. J. Optim. (2016)
Lloyd, N.G.: Degree Theory. Cambridge University Press, London (1978)
Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math. Oper. Res. 18, 868–879 (1993)
Isac, G., Bulavski, V., Kalashnikov, V.: Exceptional families, topological degree and complementarity problems. J. Global Optim. 10, 207–225 (1997)
Isac, G., Obuchowska, W.T.: Functions without EFE and complementarity problems. J. Optim. Theory Appl. 99(1), 147–163 (1998)
Zhao, Y.B., Han, J.: Exceptional family of elements for a variational inequality problem and its applications. J. Global Optim. 14, 313–330 (1999)
Zhao, Y.B., Han, J., Qi, H.D.: Exceptional families and existence theorems for variational inequality problems. J. Optim. Theory Appl. 101(2), 475–495 (1999)
Huang, Z.H.: Generalization of an existence theorem for variational inequalities. J. Optim. Theory Appl. 118(3), 567–585 (2003)
Han, J., Huang, Z.H., Fang, S.-C.: Solvability of variational inequality problems. J. Optim. Theory Appl. 122(3), 501–520 (2004)
Agangić, M., Cottle, R.W.: A note on \(Q\)-matrices. Math. Program. 16, 374–377 (1979)
Danao, R.A.: \(Q\)-matrices and boundedness of solutions to linear complementarity problems. J. Optim. Theory Appl. 83(2), 321–332 (1994)
Hu, S., Huang, Z.H., Qi, L.: Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math. 57, 181–195 (2014)
Gowda, M.S., Luo, Z., Qi, L., Xiu, N.: \(Z\)-tensors and complementarity problems. arXiv: 1510.07933v2 (2016)
Karamardian, S.: An existence theorem for the complementarity problem. J. Optim. Theory Appl. 19(2), 227–232 (1976)
Chen, H.B., Wang, Y.J.: High-order copositive tensors and its applications. J. Appl. Anal. Comput. 8(6), 1863–1885 (2018)
Chen, H.B., Huang, Z.H., Qi, L.: Copositivity detection of tensors: Theory and algorithm. J. Optim. Theory Appl. 174, 746–761 (2017)
Chen, H.B., Huang, Z.H., Qi, L.: Copositive tensor detection and its applications in physics and hypergraphs. Comput. Optim. Appl. 69, 133–158 (2018)
Li, L., Zhang, X., Huang, Z.H., Qi, L.: Test of copositive tensors. J. Ind. Manag. Optim. 15(2), 881–891 (2019)
Gowda, M.S.: Polynomial complementarity problems. Pac. J. Optim. 13(2), 227–241 (2017)
Yu, W., Ling, C., He, H.: On the properties of tensor complementarity problems. Pac. J. Optim. 14(4), 675–691 (2018)
Song, Y., Qi, L.: Strictly semi-positive tensors and the boundedness of tensor complementarity problems. Optim. Lett. 11(7), 1407–1426 (2017)
Song, Y., Mei, W.: Structural properties of tensors and complementarity problems. J. Optim. Theory Appl. 176, 289–305 (2018)
Xu, Y., Gu, W.Z., Huang, Z.H.: Estimations on upper and lower bounds of solutions to a class of tensor complementarity problems. Front. Math. China 14(3), 661–671 (2019)
Samelson, H., Thrall, R.M., Wesler, O.: A partitioning theorem for Euclidean \(n\)-space. Proc. Am. Math. Soc. 9, 805–807 (1958)
Megiddo, N., Kojima, M.: On the existence and uniqueness of solutions in nonlinear complementarity problems. Math. Program. 12, 110–130 (1977)
Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. Optim. 18, 461–481 (2007)
Miao, X.H., Huang, Z.H.: GUS-property for Lorentz cone linear complementarity problems on Hilbert spaces. Sci. China Math. 54, 1259–1268 (2011)
Wang, Y., Huang, Z.H., Qi, L.: Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177, 137–152 (2018)
Moré, J.J.: Coercivity conditions in nonlinear complementarity problems. Math. Program. 16, 1–16 (1974)
Liu, D.D., Li, W., Vong, S.W.: Tensor complementarity problems: the GUS-property and an algorithm. Linear Multilinear Algebra 66(9), 1726–1749 (2018)
Balaji, R., Palpandi, K.: Positive definite and Gram tensor complementarity problems. Optim. Lett. 12, 639–648 (2018)
Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169(3), 1069–1078 (2016)
Chen, H.B., Qi, L., Song, Y.: Column sufficient tensors and tensor complementarity problems. Front. Math. China 13(2), 255–276 (2018)
Zheng, Y.N., Wu, W.: On a class of semi-positive tensors in tensor complementarity problem. J. Optim. Theory Appl. 177, 127–136 (2018)
Zheng, M.M., Zhang, Y., Huang, Z.H.: Global error bounds for the tensor complementarity problem with a \(P\)-tensor. J. Ind. Manag. Optim. 15(2), 933–946 (2019)
Mathias, R., Pang, J.-S.: Error bounds for the linear complementarity problem with a \(P\)-matrix. Linear Algebra Appl. 132, 123–136 (1990)
Gowda, M.S.: On the continuity of the solution map in linear complementarity problems. SIAM J. Optim. 2(4), 619–634 (1992)
Gowda, M.S., Pang, J.S.: On solution stability of the linear complementarity problem. Math. Oper. Res. 17(1), 77–83 (1992)
Bai, X.L., Huang, Z.H., Li, X.: Stability of solutions and continuity of solution maps of tensor complementarity problems. Asia Pac. J. Oper. Res. 36(2), 1940002 (19 pages) (2019). https://doi.org/10.1142/S0217595919400025
Hieu, VuT: On the \(R_0\)-tensors and the solution map of tensor complementarity problems. J. Optim. Theory Appl. 181, 163–183 (2019)
Luo, Z., Qi, L., Xiu, N.: The sparsest solutions to \(Z\)-tensor complementarity problems. Optim. Lett. 11(3), 471–482 (2017)
Guo, Q., Zheng, M.M., Huang, Z.H.: Properties of \(S\)-tensors. Linear Multilinear Algebra 67(4), 685–696 (2019)
Wang, X.Y., Chen, H.B., Wang, Y.J.: Solution structures of tensor complementarity problem. Front. Math. China 13(4), 935–945 (2018)
Ling, L., He, H., Ling, C.: On error bounds of polynomial complementarity problems with structured tensors. Optimization 67(2), 341–358 (2018)
Mangasarian, O.L., Ren, J.: New error bounds for the nonlinear complementarity problem. Comm. Appl. Nonlinear Anal. 1, 49–56 (1994)
Ling, L., Ling, C., He, H.: Properties of the solution set of generalized polynomial complementarity problems. arXiv: 1905.00670 (2019)
Wang, J., Hu, S., Huang, Z.H.: Solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 176, 120–136 (2018)
Xu, Y., Gu, W.Z, Huang, H.: Solvability of two classes of tensor complementarity problems. Math. Probl. Eng. 2019, Article ID 6107517 (8 pages). https://doi.org/10.1155/2019/6107517 (2019)
Hu, S., Wang, J., Huang, Z.H.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 179, 983–1000 (2018)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 457–464 (2009)
Yang, L., Huang, Z.H., Shi, X.J.: A fixed point iterative method for low-rank tensor pursuit. IEEE Trans. Signal Process. 61(11), 2952–2962 (2013)
Tawhid, Mohamed A., Rahmati, S.: Complementarity problems over a hypermatrix (tensor) set. Optim. Lett. 12, 1443–1454 (2018)
Gowda, M.S., Sznajder, R.: General order linear complementarity problem. SIAM J. Matrix Anal. Appl. 15(3), 779–795 (1994)
Che, M., Qi, L., Wei, Y.: The generalized order tensor complementarity problems. Numer. Math. Theor. Meth. Appl. 12(4), 1–19 (2019)
Che, M., Qi, L., Wei, Y.: Stochastic \(R_0\) tensors to stochastic tensor complementarity problems. Optim. Lett. 13, 261–279 (2019)
Song, Y., Qi, L.: Eigenvalue analysis of constrained minimization problem for homogeneous polynomial. J. Global Optim. 64(3), 563–575 (2016)
Ling, C., He, H., Qi, L.: On the cone eigenvalue complementarity problem for higher-order tensors. Comput. Optim. Appl. 63, 143–168 (2016)
Ling, C., He, H., Qi, L.: Higher-degree eigenvalue complementarity problems for tensors. Comput. Optim. Appl. 64(1), 149–176 (2016)
Hou, J., Ling, C., He, H.: A class of second-order cone eigenvalue complementarity problems for higher-order tensors. J. Oper. Res. Soc. China 5(1), 45–64 (2017)
Fan, J., Nie, J., Zhou, A.: Tensor eigenvalue complementarity problems. Math. Program. 170, 507–539 (2018)
Wang, X.Y., Chen, H.B., Wang, Y.J.: On the solution existence of Cauchy tensor variational inequality problems. Pac. J. Optim. 14(3), 479–487 (2018)
Acknowledgements
We are very grateful to professors Chen Ling, Yisheng Song, Shenglong Hu and Ziyan Luo for reading the first draft of this paper and putting forward valuable suggestions for revision. The first author’s work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002 and 11871051), and the second author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717).
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Huang, ZH., Qi, L. Tensor Complementarity Problems—Part I: Basic Theory. J Optim Theory Appl 183, 1–23 (2019). https://doi.org/10.1007/s10957-019-01566-z
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DOI: https://doi.org/10.1007/s10957-019-01566-z
Keywords
- Tensor complementarity problem
- Nonemptiness and compactness of the solution set
- Global uniqueness and solvability
- Error bound theory
- Stability and continuity analysis