A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem

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  • School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received date: 2019-11-01

  Revised date: 2020-04-14

  Online published: 2021-11-25

Abstract

This paper considers the tensor split feasibility problem. Let C and Q be non-empty closed convex set and $\mathcal{A}$ be a semi-symmetric tensor. The tensor split feasibility problem is to find xC such that $\mathcal{A} x^{m-1} \in Q$. If we simply take this problem as a special case of the nonlinear split feasibility problem, then we can directly get a projection method to solve it. However, applying this kind of projection method to solve the tensor split feasibility problem is not so efficient. So we propose a Levenberg– Marquardt method to achieve higher efficiency. Theoretical analyses are conducted, and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method.

Cite this article

Yu-Xuan Jin, Jin-Ling Zhao . A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem[J]. Journal of the Operations Research Society of China, 2021 , 9(4) : 797 -817 . DOI: 10.1007/s40305-020-00337-2

References

[1] Bader, Brett W., Kolda, Tamara G. Tamara G., et.al.:MATLAB Tensor Toolbox Version 2.6, URL:http://www.sandia.gov/tgkolda/TensorToolbox/, Accessed, February 2015
[2] Byrne, C.:Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441-453(2002)
[3] Byrne, C.:A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20, 103-120(2004)
[4] Censor, Y., Elfving, T.:A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221-239(1994)
[5] Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.:A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365(2006)
[6] Censor, Y., Elfving, T., Kopf, N., et al.:The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems 21, 2071-2084(2005)
[7] Dang, Y., Gao, Y.:The strong convergence of a KMCCQ-like algorithm for a split feasibility problem. Inverse Problems 27, 015007(2011)
[8] Dang, Y.Z., Xue, Z.H., Wang, B.:Hybrid CQ projection algorithm with line-search process for the split feasibility problem. Journal of Inequalities and Applications 106, (2016). https://doi.org/10.1186/s13660-016-1039-7
[9] Dang, Y.Z., Sun, J., Xu, H.L.:Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization 13, 1383-1394(2017)
[10] Ding, W.Y., Wei, Y.M.:Solving multi-linear systems with M-tensors. J. Sci. Comput. 68, 689-715(2016)
[11] Fan, J.Y., Pan, J.Y.:A note on the Levenberg-Marquardt parameter. Appl. Math. Comput. 207, 351-359(2009)
[12] Fan, J.Y., Yuan, Y.X.:On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption. Computing 74, 23-39(2005)
[13] He, B.S., He, X.Z., Liu, H.X., Wu, T.:Self-adaptive projection method for co-coercive variational inequalities. European Journal of Operational Research 196, 43-48(2009)
[14] He, S., Tian, H., Xu, H.-K.:The selective projection method for convex feasibility and split feasibility problems. Journal of Nonlinear and Convex Analysis 19(7), 1199-1215(2018)
[15] Huang, B.H., Ma, C.F.:The modulus-based Levenberg-Marquardt method for solving linear complementarity problem. Numerical Mathematics-Theory Methods and Applications 12(1), 154-168(2018)
[16] Huang, B.H., Ma, C.F.:Accelerated modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems. Computational and Applied Mathematics 37, 3053-3076(2018)
[17] Ke, Y.F., Ma, C.F.:On the convergence analysis of two-step modulus-based matrix splitting iteration method for linear complementarity problems. Applied Mathematics and Computation 243, 413-418(2014)
[18] Levenberg, K.:A method for the solution of certain non-linear problems in least squares. Quarterly Applied Math. 2, 436-438(1994)
[19] Li, X.T., Ng, M.K.:Solving sparse non-negative tensor equations:algorithms and applications. Front. Math. China 10, 649-680(2015)
[20] Li, Y., Huang, Z., Hu, S.:Connectedness of the solution set of the tensor complementarity problem. Journal of Mathematical Analysis and Applications 487, 123965(2020)
[21] Li, Z., Han, D., Zhang, W.:A self-adaptive projection-type method for nonlinear multiple-sets split feasibility problem. Inverse Problems in Science and Engineering 21, 155-170(2013)
[22] Lv, C.Q., Ma, C.F.:A Levenberg-Marquardt method for solving semi-symmetric tensor equations[J]. Journal of Computational and Applied Mathematics 332, 13-25(2018)
[23] Ma, C.F., Huang, N.:Modified modulus-based matrix splitting algorithms for a class of weakly nondifferentiable nonlinear complementarity problems. Applied Numerical Mathematics 108, 116-124(2016)
[24] Marquardt, D.:An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431-441(1963)
[25] Qi, L.:Eigenvalues of a real supersymmetric tensor. Journal of Symbolic Computation 40, 1302-1324(2005)
[26] Wang, Y., Huang, Z.H., Qi, L.:Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177(1), 137-152(2018)
[27] Xu,H.:AvariableKrasnoselskii-Mannalgorithmandthemultiple-setsplitfeasibilityproblem.Inverse Problems 22, 2021-2034(2006)
[28] Yamashita, N., Fukushima, M.:On the rate of convergence of the Levenberg-Marquardt method. Computing Suppl. 15, 237-249(2001)
[29] Zarantonello, E. H.:Projections on convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis, ed. E. H. Zarantonello, New York:Academic (1971)
[30] Zhang, W., Han, D., Li, Z.:A self-adaptive projection method for solving the multiple-sets split feasibility problem. Inverse problems 25, 115001(2009)
[31] Zhao, J., Yang, Q.:Self-adaptive projection methods for the multiple-sets split feasibility problem. Inverse Problems 27, 035009(2011)
[32] Zhao, J., Yang, Q.:Several acceleration schemes for solving the multiple-sets split feasibility problem. Linear Algebra and its Applications 437, 1648-1657(2012)
[33] Zhao, J., Zhang, Y., Yang, Q.:Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem. Applied Mathematics and Computation 219, 1644-1653(2012)
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