Derivatives of eigenvalues and Jordan frames

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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

Abstract
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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 and 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.
[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.
[3]	J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994.
[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.
[5]	L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.
[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.
[7]	L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286.
[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.
[9]	R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143.
[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.
[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.
[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.
[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.
[14]	P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[3]	J. Faraut and A. Korányi, Analysis on Symmetric Cones, The Clarendon Press Oxford University Press, New York, 1994.
[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.
[5]	L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.
[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.
[7]	L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239 (2002), 117-129. doi: 10.1007/s002090100286.
[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.
[9]	R. A. Hauser and O. Güler, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), 121-143.
[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.
[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.
[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.
[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.
[14]	P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), 377-387.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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