American Institute of Mathematical Sciences

Advanced
2016, 6(2): 115-126. doi: 10.3934/naco.2016003

Derivatives of eigenvalues and Jordan frames

Manuel V. C. Vieira 1,

1. 

Departamento de Matemática, Faculdade de Ciências e Tecnologia & CMA, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal

Received  January 2015 Revised  May 2016 Published  June 2016

Every element in a Euclidean Jordan algebra has a spectral decomposition. This spectral decomposition is generalization of the spectral decompositions of a matrix. In the context of Euclidean Jordan algebras, this is written using eigenvalues and the so-called Jordan frame. In this paper we deduce the derivative of eigenvalues in the context of Euclidean Jordan algebras. We also deduce the derivative of the elements of a Jordan frame associated to the spectral decomposition.
Keywords: symmetric cones, differentiability., eigenvalues, Euclidean Jordan algebras.
Mathematics Subject Classification: Primary: 90C25, 17C27; Secondary: 58B1.
Citation: Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003
References:
[1]

M. Baes, Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl., 422 (2007), 664-700. doi: 10.1016/j.laa.2006.11.025.  Google Scholar

[2]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar

[3]

J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994.  Google Scholar

[4]

L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar

[5]

L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.  Google Scholar

[6]

L. Faybusovich and R. Arana, A long-step primal-dual algorithm for the symmetric programming problem, Systems Control Lett., 43 (2001), 3-7. doi: 10.1016/S0167-6911(01)00092-5.  Google Scholar

[7]

L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286.  Google Scholar

[8]

G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step infeasible interior-point method for symmetric optimization, European Journal of Operational Research, 214 (2011), 473-484. doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

[9]

R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143.  Google Scholar

[10]

R. A. Hauser and Y. Lim, Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim., 12 (2002), 715-723. doi: 10.1137/S1052623400370953.  Google Scholar

[11]

Z.-H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 450 (2010), 557-579. doi: 10.1007/s10589-008-9180-y.  Google Scholar

[12]

B. Kheirfam, A corrector-predictor path-following method for convex quadratic symmetric cone optimization, Journal of Optimization Theory and Applications, 164 (2015), 246-260. doi: 10.1007/s10957-014-0554-2.  Google Scholar

[13]

M. Kojima and M. Muramatsu, An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones, Math. Program., 110 (2007), 315-336. doi: 10.1007/s10107-006-0004-5.  Google Scholar

[14]

P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387.  Google Scholar

[15]

G. Lesaja and C. Roos, Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones, Journal of Optimization Theory and Applications, 150 (2011), 444-474. doi: 10.1007/s10957-011-9848-9.  Google Scholar

[16]

H. Liu, X. Yang and C. Liu, A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming, Journal of Optimization Theory and Applications, 158 (2013), 796-815. doi: 10.1007/s10957-013-0303-y.  Google Scholar

[17]

Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Algorithms in Convex Programming, SIAM: Studies in Applied and Numerical Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[18]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z.  Google Scholar

[19]

S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. doi: 10.1287/moor.26.3.543.10582.  Google Scholar

[20]

M. V. C. Vieira, Interior-point methods based on kernel functions for symmetric optimization, Optimization Methods and Software, 27 (2012), 513-537. doi: 10.1080/10556788.2010.544877.  Google Scholar

[21]

G. Q. Wang, C. J. Yub and K. L. Teo, A new full Nesterov-Todd step feasible interior-point method for convex quadratic symmetric cone optimization, Applied Mathematics and Computation, 221 (2013), 329-343. doi: 10.1016/j.amc.2013.06.064.  Google Scholar

[22]

Z. Yu, Y. Zhu and Q. Cao, On the convergence of central path and generalized proximal point method for symmetric cone linear programming, Applied Mathematics & Information Sciences, 7 (2103), 2327-2333. doi: 10.12785/amis/070624.  Google Scholar

show all references

References:
[1]

M. Baes, Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras, Linear Algebra Appl., 422 (2007), 664-700. doi: 10.1016/j.laa.2006.11.025.  Google Scholar

[2]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar

[3]

J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994.  Google Scholar

[4]

L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar

[5]

L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.  Google Scholar

[6]

L. Faybusovich and R. Arana, A long-step primal-dual algorithm for the symmetric programming problem, Systems Control Lett., 43 (2001), 3-7. doi: 10.1016/S0167-6911(01)00092-5.  Google Scholar

[7]

L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286.  Google Scholar

[8]

G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step infeasible interior-point method for symmetric optimization, European Journal of Operational Research, 214 (2011), 473-484. doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

[9]

R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143.  Google Scholar

[10]

R. A. Hauser and Y. Lim, Self-scaled barriers for irreducible symmetric cones, SIAM J. Optim., 12 (2002), 715-723. doi: 10.1137/S1052623400370953.  Google Scholar

[11]

Z.-H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 450 (2010), 557-579. doi: 10.1007/s10589-008-9180-y.  Google Scholar

[12]

B. Kheirfam, A corrector-predictor path-following method for convex quadratic symmetric cone optimization, Journal of Optimization Theory and Applications, 164 (2015), 246-260. doi: 10.1007/s10957-014-0554-2.  Google Scholar

[13]

M. Kojima and M. Muramatsu, An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones, Math. Program., 110 (2007), 315-336. doi: 10.1007/s10107-006-0004-5.  Google Scholar

[14]

P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387.  Google Scholar

[15]

G. Lesaja and C. Roos, Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones, Journal of Optimization Theory and Applications, 150 (2011), 444-474. doi: 10.1007/s10957-011-9848-9.  Google Scholar

[16]

H. Liu, X. Yang and C. Liu, A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming, Journal of Optimization Theory and Applications, 158 (2013), 796-815. doi: 10.1007/s10957-013-0303-y.  Google Scholar

[17]

Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Algorithms in Convex Programming, SIAM: Studies in Applied and Numerical Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[18]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z.  Google Scholar

[19]

S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. doi: 10.1287/moor.26.3.543.10582.  Google Scholar

[20]

M. V. C. Vieira, Interior-point methods based on kernel functions for symmetric optimization, Optimization Methods and Software, 27 (2012), 513-537. doi: 10.1080/10556788.2010.544877.  Google Scholar

[21]

G. Q. Wang, C. J. Yub and K. L. Teo, A new full Nesterov-Todd step feasible interior-point method for convex quadratic symmetric cone optimization, Applied Mathematics and Computation, 221 (2013), 329-343. doi: 10.1016/j.amc.2013.06.064.  Google Scholar

[22]

Z. Yu, Y. Zhu and Q. Cao, On the convergence of central path and generalized proximal point method for symmetric cone linear programming, Applied Mathematics & Information Sciences, 7 (2103), 2327-2333. doi: 10.12785/amis/070624.  Google Scholar

[1]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
[2]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
[3]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[4]

Xiao-Hong Liu, Wei Wu. Coerciveness of some merit functions over symmetric cones. Journal of Industrial & Management Optimization, 2009, 5 (3) : 603-613. doi: 10.3934/jimo.2009.5.603
[5]

Vincent Astier, Thomas Unger. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements, 2018, 25: 16-26. doi: 10.3934/era.2018.25.003
[6]

Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583
[7]

Xiao-Hong Liu, Wei-Zhe Gu. Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones. Journal of Industrial & Management Optimization, 2010, 6 (2) : 363-380. doi: 10.3934/jimo.2010.6.363
[8]

Brigitte Vallée. Euclidean dynamics. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 281-352. doi: 10.3934/dcds.2006.15.281
[9]

Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923
[10]

Víctor Ayala, Adriano Da Silva, Philippe Jouan. Jordan decomposition and the recurrent set of flows of automorphisms. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1543-1559. doi: 10.3934/dcds.2020330
[11]

A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20.

[12]

Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405
[13]

Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979
[14]

Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581
[15]

Pooja Louhan, S. K. Suneja. On fractional vector optimization over cones with support functions. Journal of Industrial & Management Optimization, 2017, 13 (2) : 549-572. doi: 10.3934/jimo.2016031
[16]

Songting Luo, Leonidas J. Guibas, Hong-Kai Zhao. Euclidean skeletons using closest points. Inverse Problems & Imaging, 2011, 5 (1) : 95-113. doi: 10.3934/ipi.2011.5.95
[17]

A. S. Dzhumadil'daev. Jordan elements and Left-Center of a Free Leibniz algebra. Electronic Research Announcements, 2011, 18: 31-49. doi: 10.3934/era.2011.18.31
[18]

Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75.

[19]

Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549
[20]

Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3525-3550. doi: 10.3934/jimo.2020131

 Impact Factor: 

Tools

Metrics

  • PDF downloads (246)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy