Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming
Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue bas...
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| Autor*in: |
Hu, Shenglong [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013 |
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| Schlagwörter: |
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| Systematik: |
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| Anmerkung: |
© Springer Science+Business Media New York 2013 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of optimization theory and applications - Springer US, 1967, 158(2013), 3 vom: 02. März, Seite 717-738 |
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| Übergeordnetes Werk: |
volume:158 ; year:2013 ; number:3 ; day:02 ; month:03 ; pages:717-738 |
| Links: |
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| DOI / URN: |
10.1007/s10957-013-0293-9 |
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OLC2026495106 |
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| 520 | |a Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. | ||
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| 650 | 4 | |a Semi-definite programming problem | |
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| 700 | 1 | |a Qi, Liqun |4 aut | |
| 700 | 1 | |a Song, Yisheng |4 aut | |
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10.1007/s10957-013-0293-9 doi (DE-627)OLC2026495106 (DE-He213)s10957-013-0293-9-p DE-627 ger DE-627 rakwb eng 330 510 000 VZ 17,1 ssgn SA 6420 VZ rvk SA 6420 VZ rvk Hu, Shenglong verfasserin aut Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. Symmetric tensors Maximum eigenvalue Sum of squares of polynomials Semi-definite programming problem Li, Guoyin aut Qi, Liqun aut Song, Yisheng aut Enthalten in Journal of optimization theory and applications Springer US, 1967 158(2013), 3 vom: 02. März, Seite 717-738 (DE-627)129973467 (DE-600)410689-1 (DE-576)015536602 0022-3239 nnns volume:158 year:2013 number:3 day:02 month:03 pages:717-738 https://doi.org/10.1007/s10957-013-0293-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_32 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 6420 SA 6420 AR 158 2013 3 02 03 717-738 |
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10.1007/s10957-013-0293-9 doi (DE-627)OLC2026495106 (DE-He213)s10957-013-0293-9-p DE-627 ger DE-627 rakwb eng 330 510 000 VZ 17,1 ssgn SA 6420 VZ rvk SA 6420 VZ rvk Hu, Shenglong verfasserin aut Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. Symmetric tensors Maximum eigenvalue Sum of squares of polynomials Semi-definite programming problem Li, Guoyin aut Qi, Liqun aut Song, Yisheng aut Enthalten in Journal of optimization theory and applications Springer US, 1967 158(2013), 3 vom: 02. März, Seite 717-738 (DE-627)129973467 (DE-600)410689-1 (DE-576)015536602 0022-3239 nnns volume:158 year:2013 number:3 day:02 month:03 pages:717-738 https://doi.org/10.1007/s10957-013-0293-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_32 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 6420 SA 6420 AR 158 2013 3 02 03 717-738 |
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10.1007/s10957-013-0293-9 doi (DE-627)OLC2026495106 (DE-He213)s10957-013-0293-9-p DE-627 ger DE-627 rakwb eng 330 510 000 VZ 17,1 ssgn SA 6420 VZ rvk SA 6420 VZ rvk Hu, Shenglong verfasserin aut Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. Symmetric tensors Maximum eigenvalue Sum of squares of polynomials Semi-definite programming problem Li, Guoyin aut Qi, Liqun aut Song, Yisheng aut Enthalten in Journal of optimization theory and applications Springer US, 1967 158(2013), 3 vom: 02. März, Seite 717-738 (DE-627)129973467 (DE-600)410689-1 (DE-576)015536602 0022-3239 nnns volume:158 year:2013 number:3 day:02 month:03 pages:717-738 https://doi.org/10.1007/s10957-013-0293-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_32 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 6420 SA 6420 AR 158 2013 3 02 03 717-738 |
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10.1007/s10957-013-0293-9 doi (DE-627)OLC2026495106 (DE-He213)s10957-013-0293-9-p DE-627 ger DE-627 rakwb eng 330 510 000 VZ 17,1 ssgn SA 6420 VZ rvk SA 6420 VZ rvk Hu, Shenglong verfasserin aut Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. Symmetric tensors Maximum eigenvalue Sum of squares of polynomials Semi-definite programming problem Li, Guoyin aut Qi, Liqun aut Song, Yisheng aut Enthalten in Journal of optimization theory and applications Springer US, 1967 158(2013), 3 vom: 02. März, Seite 717-738 (DE-627)129973467 (DE-600)410689-1 (DE-576)015536602 0022-3239 nnns volume:158 year:2013 number:3 day:02 month:03 pages:717-738 https://doi.org/10.1007/s10957-013-0293-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_32 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 6420 SA 6420 AR 158 2013 3 02 03 717-738 |
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10.1007/s10957-013-0293-9 doi (DE-627)OLC2026495106 (DE-He213)s10957-013-0293-9-p DE-627 ger DE-627 rakwb eng 330 510 000 VZ 17,1 ssgn SA 6420 VZ rvk SA 6420 VZ rvk Hu, Shenglong verfasserin aut Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2013 Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. Symmetric tensors Maximum eigenvalue Sum of squares of polynomials Semi-definite programming problem Li, Guoyin aut Qi, Liqun aut Song, Yisheng aut Enthalten in Journal of optimization theory and applications Springer US, 1967 158(2013), 3 vom: 02. März, Seite 717-738 (DE-627)129973467 (DE-600)410689-1 (DE-576)015536602 0022-3239 nnns volume:158 year:2013 number:3 day:02 month:03 pages:717-738 https://doi.org/10.1007/s10957-013-0293-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_32 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4266 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 6420 SA 6420 AR 158 2013 3 02 03 717-738 |
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Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. © Springer Science+Business Media New York 2013 |
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Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. © Springer Science+Business Media New York 2013 |
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Abstract Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results. © Springer Science+Business Media New York 2013 |
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