Some useful inequalities via trace function method in Euclidean Jordan algebras

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2014, 4(1): 39-48. doi: 10.3934/naco.2014.4.39

Some useful inequalities via trace function method in Euclidean Jordan algebras

Yu-Lin Chang ^1, and Chin-Yu Yang ^1,

1.	Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan

Received May 2013 Revised November 2013 Published December 2013

Abstract
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In this paper, we establish convexity of some functions associated with symmetric cones, called SC trace functions. As illustrated in the paper, these functions play a key role in the development of penalty and barrier function methods for symmetric cone programs. With trace function method we offer much simpler proofs to these useful inequalities.

Keywords: convexity, Symmetric cone, trace function., Löwner operator.

Mathematics Subject Classification: Primary: 26A27, 26B05, 26B35, 49J52, 90C3.

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

References:

[1]	A. Auslender, Penalty and barrier methods: a unified framework, SIAM Journal on Optimization, 10 (1999), 211-230. doi: 10.1137/S1052623497324825.
[2]	A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, Optimization Methods and Software, 18 (2003), 359-376. doi: 10.1080/1055678031000122586.
[3]	A. Auslender and H. Ramirez, Penalty and barrier methods for convex semidefinite programming, Mathematical Methods of Operations Research, 63 (2003), 195-219. doi: 10.1007/s00186-005-0054-0.
[4]	D. P. Bertsekas, Nonlinear Programming, 2nd edition,
[5]	R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.
[6]	A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, MPS-SIAM Series on Optimization. SIAM, Philadelphia, USA, 2001. doi: 10.1137/1.9780898718829.
[7]	Y.-Q. Bai and G. Q. Wang, Primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function, Acta Mathematica Sinica, 23 (2007), 2027-2042. doi: 10.1007/s10114-007-0967-z.
[8]	H. Bauschke, O. Güler, A. S. Lewis and S. Sendow, Hyperbolic polynomial and convex analysis, Canadian Journal of Mathematics, 53 (2001), 470-488. doi: 10.4153/CJM-2001-020-6.
[9]	Y.-L. Chang and J.-S. Chen, Convexity of symmetric cone trace functions in Euclidean Jordan algebras, Journal of Nonlinear and Convex Analysis, 14 (2013), 53-61.
[10]	J.-S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second-order cone, Mathmatical Programming, 101 (2004), 95-117. doi: 10.1007/s10107-004-0538-3.
[11]	J.-S. Chen, The convex and monotone functions associated with second-order cone, Optimization, 55 (2006), 363-385. doi: 10.1080/02331930600819514.
[12]	J.-S. Chen, T.-K. Liao and S.-H. Pan, Using Schur Complement Theorem to prove convexity of some SOC-functions, submitted manuscript, 2011.
[13]	M. Fukushima, Z.-Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM Journal on Optimazation, 12 (2002), 436-460. doi: 10.1137/S1052623400380365.
[14]	J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[15]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986.
[16]	A. Korányi, Monotone functions on formally real Jordan algebras, Mathematische Annalen, 269 (1984), 73-76. doi: 10.1007/BF01455996.
[17]	M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A.Brieg and S.Walcher, Springer, Berlin, 1999.
[18]	R. D. C. Monteiro and T. Tsuchiya, Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions, Mathematical Programming, 88 (2000), 61-83. doi: 10.1007/PL00011378.
[19]	J. Peng, C. Roos and T. Terlaky, Self-Regularity, A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, 2002.
[20]	R. Sznajder, M. S. Gowda and M. M. Moldovan, More results on Schur complements in Euclidean Jordan algebras, J. Glob. Optim., 53 (2012), 121-134. doi: 10.1007/s10898-011-9734-x.
[21]	D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras, Mathematics of Operations Research, 33 (2008), 421-445. doi: 10.1287/moor.1070.0300.
[22]	T. Tsuchiya, A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming, Optimization Methods and Software, 11 (1999), 141-182. doi: 10.1080/10556789908805750.

show all references

References:

[1]	A. Auslender, Penalty and barrier methods: a unified framework, SIAM Journal on Optimization, 10 (1999), 211-230. doi: 10.1137/S1052623497324825.
[2]	A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, Optimization Methods and Software, 18 (2003), 359-376. doi: 10.1080/1055678031000122586.
[3]	A. Auslender and H. Ramirez, Penalty and barrier methods for convex semidefinite programming, Mathematical Methods of Operations Research, 63 (2003), 195-219. doi: 10.1007/s00186-005-0054-0.
[4]	D. P. Bertsekas, Nonlinear Programming, 2nd edition,
[5]	R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.
[6]	A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications, MPS-SIAM Series on Optimization. SIAM, Philadelphia, USA, 2001. doi: 10.1137/1.9780898718829.
[7]	Y.-Q. Bai and G. Q. Wang, Primal-dual interior-point algorithms for second-order cone optimization based on a new parametric kernel function, Acta Mathematica Sinica, 23 (2007), 2027-2042. doi: 10.1007/s10114-007-0967-z.
[8]	H. Bauschke, O. Güler, A. S. Lewis and S. Sendow, Hyperbolic polynomial and convex analysis, Canadian Journal of Mathematics, 53 (2001), 470-488. doi: 10.4153/CJM-2001-020-6.
[9]	Y.-L. Chang and J.-S. Chen, Convexity of symmetric cone trace functions in Euclidean Jordan algebras, Journal of Nonlinear and Convex Analysis, 14 (2013), 53-61.
[10]	J.-S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second-order cone, Mathmatical Programming, 101 (2004), 95-117. doi: 10.1007/s10107-004-0538-3.
[11]	J.-S. Chen, The convex and monotone functions associated with second-order cone, Optimization, 55 (2006), 363-385. doi: 10.1080/02331930600819514.
[12]	J.-S. Chen, T.-K. Liao and S.-H. Pan, Using Schur Complement Theorem to prove convexity of some SOC-functions, submitted manuscript, 2011.
[13]	M. Fukushima, Z.-Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM Journal on Optimazation, 12 (2002), 436-460. doi: 10.1137/S1052623400380365.
[14]	J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[15]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986.
[16]	A. Korányi, Monotone functions on formally real Jordan algebras, Mathematische Annalen, 269 (1984), 73-76. doi: 10.1007/BF01455996.
[17]	M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A.Brieg and S.Walcher, Springer, Berlin, 1999.
[18]	R. D. C. Monteiro and T. Tsuchiya, Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions, Mathematical Programming, 88 (2000), 61-83. doi: 10.1007/PL00011378.
[19]	J. Peng, C. Roos and T. Terlaky, Self-Regularity, A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, 2002.
[20]	R. Sznajder, M. S. Gowda and M. M. Moldovan, More results on Schur complements in Euclidean Jordan algebras, J. Glob. Optim., 53 (2012), 121-134. doi: 10.1007/s10898-011-9734-x.
[21]	D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras, Mathematics of Operations Research, 33 (2008), 421-445. doi: 10.1287/moor.1070.0300.
[22]	T. Tsuchiya, A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming, Optimization Methods and Software, 11 (1999), 141-182. doi: 10.1080/10556789908805750.

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