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       Research       

 

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Research interests: tensors, matrices, optimization, and numerical analysis.

What is new:

TenEig - A Matlab software package for computing tensor eigenvalues and eigenvectors.

The paper Perron-Frobenius theorem for nonnegative multilinear forms and extensions (by Friedland, Gaubert, and Han, LAA, 2013) has been selected as one of the 10 Notable Papers from Linear Algebra and Its Applications published over the last 50 years.

The adaptive strategy of choosing parameters for the Nelder-Mead method proposed in Implementing the Nelder-Mead simplex algorithm with adaptive parameters (by Gao and Han, COAP, 2012) has been implemented by other researchers in various languages and packages, including R, Python, Julia, C, Mathematica, TensorFlow, MOT, Unfit,  BISTROMATH, OHANA, etc..

A Derivative Conjecture. (It is related to the characterization of the rate of convergence of iterated exponentials.)

Awards:

Received the Scholarly or Creative Achievement Award, UM-Flint, 2015.

Shared $300 monetary prize (with Jianhong Xu), which was awarded by Professor Frank Stenger for proving his Sinc matrix conjecture, 2013.

Selected Articles:

A continuation method for tensor complementarity problems, Journal of Optimization Theory and Applications, (https://link.springer.com/article/10.1007/s10957-018-1422-2), 180, 2019, pp. 949--963. 

(with L. Chen, H. Yin, and L. Zhou) A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor, Journal of Computational and Applied Mathematics, 355, 2019, pp. 174--181. Preprint. https://www.sciencedirect.com/science/article/pii/S0377042719300172

(with L. Chen and L. Zhou) Linear homotopy method for computing generalized tensor eigenpairs, Frontiers of Mathematics in China,(https://link.springer.com/article/10.1007/s11464-017-0662-z) 2017. Preprint.        

A homotopy method for solving multilinear systems with M-tensors, Applied Mathematics Letters, 69, 2017, pp.49-54.  

(http://dx.doi.org/10.1016/j.aml.2017.01.019)   Preprint.                                                                                     

(with L. Chen and L. Zhou) Computing tensor eigenvalues via homotopy methods, SIAM Journal on Matrix Analysis and Applications, 37(1), 2016, pp.290-319. (http://dx.doi.org/10.1137/15M1010725)  arXiv preprint.

(with J. Xu) Proof of Stenger's conjecture on matrix $I^{(-1)}$ of Sinc methods, Journal of Computational and Applied Mathematics, Vol. 255, 2014, pp. 805-811. (doi: 10.1016/j.cam.2013.07.001.)   Preprint. (This paper has been awarded $300 monetary prize by Professor Frank Stenger for proving his Sinc matrix conjecture.)

(with D. Bunker and S. Zhang) A proximal ANLS algorithm for nonnegative tensor factorization with a periodic enhanced line search, Applications of Mathematics, 58(5), 2013, pp.493-509.( doi:10.1007/s10492-013-0026-2.) Preprint.

An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors, Numerical Algebra, Control, and Optimization, Vol. 3, No. 3, 2013, pp.583-599.(doi:10.3934/naco.2013.3.583.)  Preprint.

(with S. Friedland and S. Gaubert) Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, Vol. 438, No. 2, 2013, pp.738-749.( doi:10.1016/j.laa.2011.02.042.) Preprint. (This paper has been selected as one of the 10 Notable Papers from Linear Algebra and Its Applications published over the last 50 years, Link.)

(with F. Gao and K. Schilling) On the rate of convergence of iterated exponentials, Journal of Applied Mathematics and Computing, Vol. 39, 2012, pp.89-96.doi: 10.1007/s12190-011-0511-2 

(with F. Gao) Implementing the Nelder-Mead simplex algorithm with adaptive parameters, Computational Optimization and Applications, Vol. 51, No.1, 2012, pp. 259-277. doi: 10.1007/s10589-010-9329-3 ( Matlab code download: here )

(with M. Neumann and U. Prasad) Alternating projected Barzilai-Borwein methods for nonnegative matrix factorization, Electronic Transactions on Numerical Analysis, Vol. 36, 2009-2010, pp. 54-82. ( Matlab code download: here )

(with J. Han, M. Neumann, U. Prasad) On the convergence of the Image Space Reconstruction Algorithm, Operators and Matrices, Vol. 3, No. 1, 2009, pp. 41-58.

(with J. Axtell, D. Hershkowitz, M. Neumann, N. Sze) Optimization of the spectral radius of a product for nonnegative matrices, Linear Algebra and its Applications, Vol. 430, Issues 5-6, 2009, pp. 1442-1451.

(with M. Neumann) Inner product spaces, orthogonal projection, least squares, and singular value decomposition, in Handbook of Linear Algebra, (editor: L. Hogben), CRC Press, 1st ed., 2006; 2nd ed., 2013.

(with M. Chen and M. Neumann) On single and double Soules matrices, Linear Algebra and its Applications, Vol. 416, 2006, pp.88-110.

(with M. Neumann) Effect of dimensionality on the Nelder-Mead simplex method, Optimization Methods and Software, Vol. 21, No. 1, 2006, pp. 1-16.

(with M. Catral, M. Neumann, and R. Plemmons) On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices, Linear Algebra and its Applications, Vol. 393, 2004, pp.107-126.

(with G. Liu) On the convergence of the UOBYQA method, Journal of Applied Mathematics and Computing, Vol. 16, No. 1-2, 2004, pp.125-142.

(with M. Neumann) Combining quasi-Newton and Cauchy directions, International Journal of Applied Mathematics, Vol. 12, No.2, 2003, pp.167-191.

(with M. Neumann and J. Xu) On the roots of certain polynomials arising from the analysis of the Nelder-Mead simplex method, Linear Algebra and its Applications, Vol. 363, 2003, pp.109-124.

(with G. Liu and L. Jing) Convergence properties of the DFP algorithm for unconstrained optimization, Optimization, Vol. 51, No. 2, 2002, pp.731-758.

(with G. Liu) A study of the Dennis-Wolkowicz method on convex problems, Computational Optimization and Applications, Vol. 19, 2001, pp.297-317.

(with G. Liu and L. Jing) A non-monotone Broyden method for unconstrained optimization, Optimization Methods and Software, Vol. 15, No. 2, 2001, pp.153-172.

(with M. Neumann and M. Tsatsomeros) Spectral radii of fixed Frobenius norm perturbations of nonnegative matrices, SIAM Journal on Matrix Analysis and Applications, Vol.21, No.1, 2000, pp.79-92.

(with G. Liu, L. Jing, and D. Han) A class of non-monotone conjugate gradient methods for unconstrained optimization, Journal of Optimization Theory and Applications, Vol.101, No.1, 1999, pp.127-140.

(with G. Liu) Global analysis of the Dennis-Wolkowicz least-change secant algorithm, SIAM Journal on Optimization, Vol.8, No.3, 1998, pp.813-832.

(with L. Elsner, I. Koltracht, M. Neumann, and M. Zippin) On a polygon equality problem, Journal of Mathematical Analysis and Applications, Vol.223, 1998, pp.67-75.

(with D. Han) Convergence analysis of the self-dual optimally conditioned SSVM method of Oren-Spedicato, Optimization, Vol.37, 1996, pp.153-169.

(with G. Liu) Global convergence of quasi-Newton methods for unconstrained optimization, Chinese Science Bulletin, No.7, 1996, pp.529-533.

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