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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. P. Bertsekas, Nonlinear Programming,, 2nd edition, ().   Google Scholar

[5]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

J.-S. Chen, The convex and monotone functions associated with second-order cone, Optimization, 55 (2006), 363-385. doi: 10.1080/02331930600819514.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

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

[15]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986.  Google Scholar

[16]

A. Korányi, Monotone functions on formally real Jordan algebras, Mathematische Annalen, 269 (1984), 73-76. doi: 10.1007/BF01455996.  Google Scholar

[17]

M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A.Brieg and S.Walcher, Springer, Berlin, 1999.  Google Scholar

[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.  Google Scholar

[19]

J. Peng, C. Roos and T. Terlaky, Self-Regularity, A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. P. Bertsekas, Nonlinear Programming,, 2nd edition, ().   Google Scholar

[5]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

J.-S. Chen, The convex and monotone functions associated with second-order cone, Optimization, 55 (2006), 363-385. doi: 10.1080/02331930600819514.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[14]

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

[15]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986.  Google Scholar

[16]

A. Korányi, Monotone functions on formally real Jordan algebras, Mathematische Annalen, 269 (1984), 73-76. doi: 10.1007/BF01455996.  Google Scholar

[17]

M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A.Brieg and S.Walcher, Springer, Berlin, 1999.  Google Scholar

[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.  Google Scholar

[19]

J. Peng, C. Roos and T. Terlaky, Self-Regularity, A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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