### AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

Yuting Chen1, Mingyuan Cao2, Yueting Yang2, Qingdao Huang3

1. 1. School of Mathematics, Jilin University, Changchun 130012, China;
2. School of Mathematics and Statistics, Beihua University, Jilin 132013, China;
3. School of Mathematics, Jilin University, Changchun 130012, China
• Received:2019-01-29 Revised:2019-01-29 Online:2021-05-15 Published:2021-04-12
• Contact: Qingdao Huang,Email:huangqd@jlu.edu.cn
• Supported by:
The research of Yueting Yang is supported in part by the NNSF of China (11171003), the Innovation Talent Training Program of Science and Technology of Jilin Province of China (20180519011JH) and the Science and Technology Development Project Program of Jilin Province (20190303132SF). The research of Mingyuan Cao is partially supported by the Project of Education Department of Jilin Province (JJKH20200028KJ). The research of Qingdao Huang is partially supported by the NNSF of China (11171131).

Yuting Chen, Mingyuan Cao, Yueting Yang, Qingdao Huang. AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS[J]. Journal of Computational Mathematics, 2021, 39(3): 358-374.

For symmetric tensors, computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere. In this paper, we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors. One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance. The other one is that a projection scheme is used to ensure the feasibility of all iteratives. Global convergence and local quadratic convergence of our algorithm are established, respectively. The preliminary numerical results show the efficiency of the proposed algorithm.

CLC Number:

 [1] L.H. Lim, Singular values and eigenvalues of tensors:a variational approach, in Computational Advances in Multi-Sensor Adaptive Processing, 20051st IEEE International Workshop on, 2005, pp. 129-132.[2] L.Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40:6(2005), 1302-1324.[3] K.C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl., 350:1(2009), 416-422.[4] G.Y. Li, L.Q. Qi and G.H. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numer. Linear. Algebr., 20:6(2013), 1001-1029.[5] Q. Ni and L.Q. Qi, A quadratically convergent algorithm for finding the largesteigenvalue of a nonnegative homogeneous polynomial map, J. Glob. Optim., 61:4(2015), 627-641.[6] J.S. Xie and A. Chang, H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph, Front. Math. China., 8:1(2013), 107-127.[7] J.S. Xie and A. Chang, On the Z-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph, Numer. Linear. Algebr., 20:6(2013), 1030-1045.[8] F. Zhang, B.Y. Zhou, L.Z. Peng, Gradient skewness tensors and local illumination detection for images, J. Comput. Appl. Math., 237:1(2013), 663-671.[9] J.M. Papy, L. De Lathauwer and S. Van Huffel, Exponential data fitting using multilinear algebra:The single-channel and multi-channel case, Numer. Linear. Algebr., 12:8(2005), 809-826.[10] J.M. Papy, L. De Lathauwer and S. Van Huffel, Exponential data fitting using multilinear algebra:The decimative case, J. Chemometr., 23:7-8(2009), 341-351.[11] L.Q. Qi and K.L. Teo, Multivariate polynomial minimization and its application in signal processing, J. Glob. Optim., 26:4(2003), 419-433.[12] T.C. Wei and P.M. Goldbart, Geometric mearsure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A., 68:1(2003), 042307.[13] Q. Ni, L.Q. Qi and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE. T. Automat. Contr., 53:5(2008), 1096-1107.[14] J.F. Cardoso, High-order contrasts for independent component analysis, Neural. Comput., 11:1(1999), 157-192.[15] L. Lathauwer, P. Comon, B. Moor and J. Vandewalle, High-order power method-application in independent component analysis, in Proceeding of the International Symposium on Nonlinear Theory and its Applications (NOLTA'95), Las Vegas, Nevada, 1995, pp. 91-96.[16] L. De Lathauwer, B. De Moor and J. Vandewalle, On the best rank-1 and rank-(R1, R2,..., RN) approximation of higher-order tensors, SIAM. J. Matrix. Anal. A., 21:4(2000), 1324-1342.[17] E. Kofidis and P.A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM. J. Matrix. Anal. A., 23:3(2002), 863-884.[18] C.L. Hao, C.F. Cui and Y.H. Dai, A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensors, Numer. Linear. Algebr., 22:2(2015), 283-298.[19] C.L. Hao, C.F. Cui and Y.H. Dai, A feasible trust-region method for calculating extreme Z-eigenvalues of symmetric tensors, Pac. J. Optim., 11:2(2015), 291-307.[20] L.Q. Qi, F. Wang and Y.J. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118:2(2009), 301-316.[21] G.H. Yu, Z.F. Yu, Y. Xu, Y.S. Song and Y. Zhou, An adaptive gradient method for computing generalized tensor eigenpairs, Comput. Optim. Appl., 65:3(2016), 781-797.[22] J. Nocedal and S. Wright, Numerical Optimization, Springer, New York, 1999.[23] Z.C. Cui and B.Y. Wu, A new modified nomonotone adaptive trust region method for unconstrained optimization, Comput. Optim. Appl., 53:3(2012), 795-806.[24] G.D. Li, Trust region method with automatic determination of the trust region radius, Chinese Journal of Engineering Mathematics., 23:5(2006), 843-848. (in Chinese)[25] S. Wang, M. Wu and Z. Jia, Matrix Inequality, Science Press, 2ed, 2006. (in Chinese)[26] M.Y. Cao, Q.D. Huang and Y.T. Yang, A self-adaptive trust region method for extreme B-eigenvalues of symmetric tensors. Numer. Algorithms., 81:2(2019), 407-420.[27] J.W. Nie and L. Wang, Semidefinite relaxations for best rank-1 tensor approximations, SIAM. J. Matrix. Anal. A., 35:3(2014), 1155-1179.[28] E.D. Dolan and J.J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91:2(2002), 201-213.
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