Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras

Linear Algebra and Its Applications

Volumn 422, Issue 2-3, 2007, Pages 664-700

  Baes, Michel ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Convexity; Differentiability; Euclidean Jordan algebras; Spectral functions

Indexed keywords

COMPUTATIONAL COMPLEXITY; EIGENVALUES AND EIGENFUNCTIONS; LINEAR ALGEBRA; MATRIX ALGEBRA; PROBLEM SOLVING;

CONVEXITY; DIFFERENTIABILITY; EUCLIDEAN JORDAN ALGEBRAS; SPECTRAL FUNCTIONS;

FUNCTION EVALUATION;

EID: 33847261677     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2006.11.025     Document Type: Article

References (35)

1. Alizadeh F., and Schmieta S. Symmetric cones, potential reduction methods and word-by-word extensions. In: Wolkowicz H., Saigal R., and Vandenberghe L. (Eds). Handbook of Semidefinite Programming (2000), Kluwer 195-223
2. M. Baes, Spectral functions on Jordan algebras: differentiability and convexity properties, CORE Discussion Paper 16, Center for Operation Research and Econometrics, Université catholique de Louvain, 2004.
3. Ben-Tal A., and Nemirovski A. Non-Euclidean restricted memory level method for large-scale convex optimization. Math. Prog. 102 (2005) 407-456
4. Braun H., and Koecher M. Jordan Algebren (1966), Springer, Berlin
5. Chen C., and Mangasarian O. Smoothing methods for convex inequalities and linear complementarity problems. Math. Prog. 71 (1995) 51-69
6. Fan K. On a theorem of Weyl concerning eigenvalues of linear transformations I. Proc. Amer. Math. Soc. 35 (1949) 652-655
7. Faraut J., and Koranyi A. Analysis on Symmetric Cones (1994), Oxford University Press, London
8. Faybusovich L. Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86 (1997) 149-175
9. Faybusovich L. A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 239 (2002) 117-129
10. L. Faybusovich, Jordan-algebraic approach to convexity theorems for quadratic mappings, Research report, University of Notre Dame, June 2005.
11. Fukushima M., Luo Z.-Q., and Tseng P. Smoothing functions for second-order-cone complementarity problems. SIAM J. Opt. 12 (2001) 436-460
12. Gowda M., Sznajder R., and Tao J. Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393 (2004) 203-232
13. Hanche-Olsen H., and Stormer E. Jordan Operator Algebras (1984), Pitman, Boston
14. Hiriart-Urruty J.-B., and Lemaréchal C. Convex Analysis and Minimization Algorithms II (1993), Springer, Heidelberg
15. Horn R., and Johnson Ch. Matrix Analysis (1996), Cambridge University Press, Cambridge
16. Koecher M. The Minnesota Notes on Jordan Algebras and Their Applications (1999), Springer, Berlin
17. M. Kojima, M. Muramatsu, An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones, Math. Prog., in press, doi:10.1007/s10107-006-0004-5.
18. Lewis A. Convex analysis on the Hermitian matrices. SIAM J. Opt. 6 (1996) 164-177
19. Lewis A. Derivatives of spectral functions. Math. Oper. Res. 21 (1996) 576-588
20. Lewis A., and Sendov H. Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23 (2002) 368-386
21. Lim Y., Kim J., and Faybusovich L. Simultaneous diagonalization on simple Euclidean Jordan algebras and its applications. Forum Mathematicum 15 (2003) 639-644
22. Malick J., and Sendov H. Clarke generalized Jacobian of the projection onto the cone of positive semidefinite matrices. Set-Valued Anal. 14 (2006) 273-293
23. Marshall A., and Olkin I. Inequalities: Theory of Majorization and its Applications (1979), Academic Press, New York
24. Massam H., and Neher E. Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras. Ann. Stat. 26 (1998) 1051-1082
25. Muramatsu M. On commutative class of search directions for linear programming over symmetric cones. J. Opt. Theor. Appl. 112 (2002) 595-625
26. Nesterov Y. Introductory Lectures on Convex Optimization: A Basic Course (2003), Kluwer, Dordrecht
27. Nesterov Y. Smooth minimization of non-smooth functions. Math. Prog. 103 (2005) 127-152
28. Overton M., and Womersley R. Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Prog. 62 (1993) 321-357
29. Peng J., Roos C., and Terlaky T. Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms (2002), Princeton University Press, Princeton
30. Rangarajan B. Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM J. Opt. 16 (2006) 1211-1229
31. Rockafellar R.T. Convex Analysis (1970), Princeton University Press, Princeton
32. Schmieta S., and Alizadeh F. Extension of commutative class of interior point algorithms to symmetric cones. Math. Prog. 96 (2003) 409-438
33. H. Sendov, Variational spectral analysis, Ph.D. thesis, University of Waterloo, Ontario, Canada, 2000.
34. D. Sun, J. Sun, Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras, Technical report, Department of Mathematics, University of Singapore, 2004.
35. Wolsey L. Integer Programming (1998), Wiley, New York