American Institute of Mathematical Sciences

Advanced
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

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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