A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming

Mathematical Programming, Series B

Volumn 81, Issue 3, 1998, Pages 281-299

  Monteiro, Renato D C ^a   Zhang, Yin ^b  

^a Georgia Institute of Technology   (United States)

^b RICE UNIVERSITY   (United States)

Author keywords

Interior point methods; Path following; Primal dual; Semidefinite programming

Indexed keywords

EID: 0001343365     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01580085     Document Type: Article

References (27)

1. F. Alizadeh, Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices, Ph.D. Thesis, Computer Science Department, University of Minnesota, Minneapolis, 1991.
2. F. Alizadeh, J.A. Haeberly, M. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, Technical report, Computer Science Department, New York University, New York, 1996.
3. C. Helmberg, F. Rendl, R.J. Vanderbei, H. Wolkowicz, An interior-point method for semidefinite programming, SIAM Journal on Optimization 6 (1996) 342-361.
4. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
5. F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM Journal on Control and Optimization 31 (1993) 1360-1377.
6. M. Kojima, S. Mizuno, A. Yoshise, A primal-dual interior point method for linear programming, in: Nimrod Megiddo (Ed.), Progress in Mathematical Programming, Interior-Point and Related Methods, Springer, New York, 1989, pp. 29-47.
7. M. Kojima, M. Shida, S. Shindoh, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs. Mathematical Programming 80 (1998) 129-160.
8. M. Kojima, M. Shida, S. Shindoh, A note on the Nesterov-Todd and the Kojima-Shindoh-Hara search directions in semidefinite programming, Research Report #B-313, Department of Mathematical and Computing Sciences. Tokyo Institute of Technology 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan, 1996.
9. M. Kojima, S. Shindoh, S. Hara, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM Journal on Optimization 7 (1997) 86-125.
10. C-J. Lin, R. Saigal, A predictor-corrector method for semi-definite programming, Working paper, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2177, 1995.
11. Z-Q. Luo, J.F. Sturm, S. Zhang, Superlinear convergence of a symmetric primal-dual path-following algorithms for semidefinite programming, Working paper, 1996 (to appear in SIAM Journal on Optimization).
12. R.D.C. Monteiro, Primal-dual path-following algorithms for semidefinite programming, SIAM Journal on Optimization 7 (1997) 663-678.
13. R.D.C. Monteiro, Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions, Manuscript, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, 1996.
14. Y.E. Nesterov, A.S. Nemirovskii, A general approach to the design of optimal methods for smooth convex functions minimization, Ekonomika i Matem, Metody 24 (1988) 509-517 (In Russian; English transl. Matekon: Translations of Russian and East European Math. Economics.).
15. Y.E. Nesterov, A.S. Nemirovskii, Self-concordant functions and polynomial time methods in convex programming, Preprint, Central Economic & Mathematical Institute, USSR Acadamy of Science, Moscow, USSR, 1989.
16. Y.E. Nesterov, A.S. Nemirovskii, Optimization over positive semidefinite matrices: Mathematical background and user's manual, Technical report, Central Economic & Mathematical Institute, USSR Academy of Science Moscow, USSR, 1990.
17. Y.E. Nesterov, A.S. Nemirovskii, Interior Point Methods in Convex Programming: Theory and Applications, Society for Industrial and Applied Mathematics, Philadelphia, 1994.
18. Y.E. Nesterov, M.J. Todd, Self-scaled barriers and interior-point methods for convex programming, Technical Report 1091, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 14853-3801, 1995.
19. Y.E. Nesterov, M.J. Todd, Primal-dual interior-point methods for self-scaled cones, Mathematics of Operations Research 22 (1997) 1-42.
20. F.A. Potra, R. Sheng, A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming, Reports on Computational Mathematics, No. 78, Department of Mathematics, The University of Iowa, 1995.
21. M. Shida, S. Shindoh, M. Kojima, Existence of search directions in interior-point algorithms for the SDP and monotone SDLCP, Technical Report B-310, Department of Mathematical and Computer Science, Tokyo Institute of Technology, 1996 (to appear in SIAM Journal on Optimization).
22. J.F. Sturm, S. Zhang, Symmetric primal-dual path-following algorithms for semidefinite programming, Report 9554/A, Ecometric Institute, Erasmus University Rotterdam, The Netherlands, 1995.
23. J.F. Sturm, S. Zhang, On the long-step path-following method for semidefinite programming, Report 9638/A, Ecometric Institute, Erasmus University Rotterdam, The Netherlands, 1996.
24. M.J. Todd, K.C. Toh, R.H. Tütüncü, On the Nesterov-Todd direction in semidefinite programming, Technical Report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA, 1996.
25. L. Vandenberghe, S. Boyd, A primal-dual potential reduction method for problems involving matrix inequalities, Mathematical Programming 69 (1995) 205-236.
26. Y. Ye, A class of projective transformations for linear programming, SIAM Journal on Computing 19 (1990) 457-466.
27. Y. Zhang, On extending some primal-dual interior-Point algorithms from linear programming to semidefinite programming, Technical Report, 1995 (to appear in SIAM Journal on Optimization).