Polynomial convergence of a new family of primal-dual algorithms for semidefinite programming

SIAM Journal on Optimization

Volumn 9, Issue 3, 1999, Pages 551-577

  Monteiro, Renato D C ^a   Tsuchiya, Takashi ^b  

^a Georgia Institute of Technology   (United States)

^b INSTITUTE OF STATISTICAL MATHEMATICS   (Japan)

Author keywords

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

Indexed keywords

EID: 0033421616     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623496312836     Document Type: Article

References (39)

1. F. ALIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. F. ALIZADEH, J.-P. HAEBERLY, AND M. OVERTON, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM J. Optim., 8 (1998), pp. 746-768.
3. R. M. FREUND, Complexity of an Algorithm for Finding an Approximate Solution of a Semidefinite Program with No Regularity Condition, Working paper OR 302-94, Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA, December 1994.
4. C. HELMBERG, F. RENDL, R. J. VANDERBEI, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim, 6 (1996), pp. 342-361.
5. F. JARRE, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control. Optim., 31 (1993), pp. 1360-1377.
6. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A polynomial-time algorithm for a class of linear complementarity problems, Math. Programming, 44 (1989), pp. 1-26.
7. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddo, ed., Springer-Verlag, New York, 1989, pp. 29-47.
8. M. KOJIMA, M. SHIDA, AND S. SHINDOH, A Note on the Nesterov-Todd and the Kojima-Shindoh-Hara Search Directions in Semidefinite Programming, Optim. Methods Softw., to appear.
9. M. KOJIMA, M. SHIDA, AND S. SHINDOH, A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, SIAM J. Optim., 9 (1999), pp. 444-465.
10. M. KOJIMA, M. SHIDA, AND S. SHINDOH, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Programming, 80 (1998), pp. 129-160.
11. M. KOJIMA, S. SHINDOH, AND S. HARA, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM J. Optim., 7 (1997), pp. 86-125.
12. C.-J. LIN AND R. SAIGAL, A Predictor-Corrector Method for Semi-Definite Programming, Working paper, Dept. of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, 1995.
13. Z.-Q. LUO, J. F. STURM, AND S. ZHANG, Superlinear convergence of a symmetric primal-dual path-following algorithm for semidefinite programming, SIAM J. Optim., 8 (1998), pp. 59-81.
14. S. MIZUNO, M. J. TODD, AND Y. YE, On adaptive step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18 (1993), pp. 945-981.
15. R. D. C. MONTEIRO, Primal-dual path-following algorithms for semidefinite programming, SIAM J. Optim., 7 (1997), pp. 663-678.
16. R. D. C. MONTEIRO, Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions, SIAM J. Optim., 8 (1998), pp. 59-81.
17. R. D. C. MONTEIRO AND I. ADLER, Interior path-following primal-dual algorithms. Part I: Linear programming, Math. Programming, 44 (1989), pp. 27-41.
18. R. D. C. MONTEIRO AND I. ADLER, Interior path-following primal-dual algorithms. Part II: Convex quadratic programming, Math. Programming, 44 (1989), pp. 43-66.
19. R. D. C. MONTEIRO AND T. TSUCHIYA, Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions, Manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, 1996; Math. Programming, to appear.
20. R. D. C. MONTEIRO AND P. R. ZANJÁCOMO, Implementation of Primal-Dual Methods for Semidefinite Programming Based on Monteiro and Tsuchiya Newton Directions and Their Variants, Tech. Report, School of ISyE, Georgia Institute of Technology, Atlanta, 1997; Optim. Math. Softw., to appear.
21. R. D. C. MONTEIRO AND Y. ZHANG, A unified analysis for a class of path-following primal-dual interior-point algorithms for semidefinite programming, Math. Programming, 81 (1998), pp. 281-299.
22. Y. E. NESTEROV AND A. S. NEMIROVSKII, A general approach to the design of optimal methods for smooth convex functions minimization, Ekonomika i Matem. Metody, 24 (1988), pp. 509-517 (in Russian). (English transl.: Matekon: Translations of Russian and East European Math. Economics.)
23. Y. E. NESTEROV AND A. S. NEMIROVSKII, A general approach to the design of optimal methods for smooth convex functions minimization, Ekonomika i Matem. Metody, 24 (1988), pp. 509-517 (in Russian). (English transl.: Matekon: Translations of Russian and East European Math. Economics.)
24. Y. E. NESTEROV AND A. S. NEMIROVSKII, Self-Concordant Functions and Polynomial Time Methods in Convex Programming, Preprint, Central Economic & Mathematical Institute, USSR Acad. Sci. Moscow, USSR, 1989.
25. Y. E. NESTEROV AND A. S. NEMIROVSKII, Optimization over Positive Semidefinite Matrices: Mathematical Background and User's Manual, Tech. report, Central Economic & Mathematical Institute, USSR Acad. Sci. Moscow, USSR, 1990.
26. Y. E. NESTEROV AND A. S. NEMIROVSKII, Interior Point Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994.
27. Y. E. NESTEROV AND M. TODD, Primal-dual interior-point methods for self-scaled cones, SIAM J. Optim., 8 (1998), pp. 324-364.
28. Y. E. NESTEROV AND M. TODD, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res., 22 (1997), pp. 1-42.
29. F. A. POTRA AND R. SHENG, On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming, Reports on Computational Mathematics 98, Dept. of Mathematics, The University of Iowa, Iowa City, 1997.
30. F. A. POTRA AND R. SHENG, A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming, SIAM J. Optim., 8 (1998), pp. 1007-1028.
31. J. F. STURM AND S. ZHANG, Symmetric Primal-Dual Path-Following Algorithms for Semidefinite Programming, Report 9554/A, Econometric Institute, Erasmus University, Rotterdam, the Netherlands, November 1995.
32. K. TANABE, Complementarity-enforcing centered Newton method for mathematical programming: Global methods, in New Method for Linear Programming, K. Tone, ed., Cooperative Research Report 5, Institute of Statistical Mathematics, Tokyo, Japan, 1987, pp. 118-144.
33. K. TANABE, Centered Newton method for mathematical programming, in System Modelling and Optimization, M. Iri and K. Yajima, eds., Springer-Verlag, Tokyo, Japan, 1988, pp. 197-206.
34. M. TODD, On the Search Directions in Interior-Point Methods for Semidefinite Programming, Tech. Report 1205, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, October, 1997.
35. M. J. TODD, K. C. TOH, AND R. H. TÜTÜNCÜ, On the Nesterov-Todd direction in semidefinite programming, SIAM J. Optim., 8 (1998), pp. 769-796.
36. P. TSENG, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, Optim. Methods Softw., 9 (1998), pp. 245-268.
37. L. VANDENBERGHE AND S. BOYD, A primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, 69 (1995), pp. 205-236.
38. Y. YE, A class of projective transformations for linear programming, SIAM J. Comput., 19 (1990), pp. 457-466.
39. Y. ZHANG, On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM J. Optim., 8 (1998), pp. 365-386.