Similarity and other spectral relations for symmetric cones

Linear Algebra and Its Applications

Volumn 312, Issue 1-3, 2000, Pages 135-154

  Sturm, Jos F ^a  

^a MAASTRICHT UNIVERSITY   (Netherlands)

Author keywords

15A48; 17C99; 90C22; 90C51; Euclidean Jordan algebra; Optimization; Symmetric cone

Indexed keywords

EID: 0034421618     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(00)00096-3     Document Type: Article

References (31)

1. F. Alizadeh, J.A. Haeberly, M.V. Nayakkankuppann, M. Overton, S. Schmieta, SDP - Pack User's Guide, New York University, New York, USA, 1997.
2. Alizadeh F., Haeberly J.A., Overton M. Complementarity and nondegeneracy in semidefinite programming. Mathematical Programming. 77(2):1997;111-128.
3. Alizadeh F., Haeberly J.A., Overton M. Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM Journal on Optimization. 8(3):1998;746-768.
4. Ando T. Concavity of certain maps and positive definite matrices and applications to Hadamard products. Linear Algebra and its Applications. 26:1979;203-241.
5. Berkelaar A.B., Sturm J.F., Zhang S. Polynomial primal-dual cone affine scaling for semidefinite programming. Applied Numerical Mathematics. 29:1999;317-333.
6. Braun H., Koecher M. Jordan Algebren. 1966;Springer, New York.
7. Faraut J., Korányi A. Analysis on Symmetric Cones: Oxford Mathematical Monographs. 1994;Oxford University Press, New York.
8. L. Faybusovich, Jordan algebras, symmetric cones and interior point methods, Technical Report, Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA, 1995.
9. L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, in: Positivity, vol. 1, Kluwer Academic Publishers, Dordrecht, 1997, pp. 331-357.
10. Faybusovich L. Linear systems in Jordan algebras and primal-dual interior-point algorithms. Journal of Computational and Applied Mathematics. 86:1997;149-175.
11. L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Technical Report, Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA, 1998.
12. Güler O. Barrier functions in interior point methods. Mathematics of Operations Research. 21:1996;860-885.
13. Güler O., Tunçel L. Characterization of the barrier parameter of homogeneous convex cones. Mathematical Programming. 81:1998;55-76.
14. Jordan P., Von Neumann J., Wigner E. On algebraic generalization of the quantum mechanical formalism. Annals of Mathematics. 36:1934;29-64.
15. n American Journal of Mathematics. 79:1957;575-596.
16. Kojima M., Megiddo N., Noma T., Yoshise A. A Unified Approach to Interior Point Algorithms for Linear Complementarity problems. 1991;Springer, Berlin.
17. Monteiro R.D.C. Primal-dual path following algorithms for semidefinite programming. SIAM Journal on Optimization. 7(3):1997;663-678.
18. Nesterov Y., Todd M.J. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research. 22(1):1997;1-42.
19. Nesterov Y., Todd M.J. Primal-dual interior-point methods for self-scaled cones. SIAM Journal on Optimization. 8:1998;324-364.
20. J.M. Ortega, Matrix Theory: A Second Course, The University Series in Mathematics, Plenum Press, New York, 1987.
21. S.H. Schmieta, F. Alizadeh, Associative algebras, symmetric cones and polynomial time interior point algorithms, Technical report, RUTCOR, Rutgers University, NJ, USA, June 1998.
22. J.F. Sturm, Primal-Dual Interior Point Approach to Semidefinite Programming, Tinbergen Institute Research Series, vol. 156, Thesis Publishers, Amsterdam, The Netherlands, 1997.
23. J.F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, in: Interior Point Methods (CD supplement with software) (special issue), Optimization Methods and Software 11-12 (1999) 625-653.
24. Sturm J.F., Zhang S. Symmetric primal-dual path following algorithms for semidefinite programming. Applied Numerical Mathematics. 29:1999;301-315.
25. Todd M.J. A study of search directions in interior-point methods for semidefinite programming. Optimization Methods and Software. 11-12:1999;1-46.
26. Todd M.J., Toh K.C., Tütüncü R.H. On the Nesterov-Todd direction in semidefinite programming. SIAM Journal on Optimization. 8(3):1998;769-796.
27. K.C. Toh, M.J. Todd, R.H. Tütüncü, SDPT3 - a MATLAB package for semidefinite programming, version 1.3, in: Interior Point Methods (CD supplement with software) (special issue), Optimization Methods and Software 11-12 (1999) 545-581.
28. T. Tsuchiya, A convergence analysis of the scaling-invariant primal-dual path-following algorithm for second-order cone programming, in: Interior Point Methods (CD supplement with software) (special issue), Optimization Methods and Software 11-12 (1999) 141-182.
29. L. Tunçel, Convex optimization: barrier functions and interior-point methods, Technical Report B-336, Department of Information Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan, 1998.
30. Tunçel L. Primal-dual symmetry and scale invariance of interior point algorithms for convex optimization. Mathematics of Operations Research. 23:1998.
31. Xue G., Ye Y. An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM Journal on Optimization. 7(4):1997;1017-1036.