A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

Mathematical Programming, Series B

Volumn 95, Issue 2, 2003, Pages 329-357

  Burer, Samuel ^a   Monteiro, Renato D C ^b  

^a UNIVERSITY OF IOWA   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Augmented lagrangian; Limited memory BFGS; Low rank factorization; Nonlinear programming; Semidefinite programming

Indexed keywords

AUGMENTED LAGRANGIAN; LIMITED MEMORY BFGS; LOW-RANK FACTORIZATION; SEMIDEFINITE PROGRAMMING;

ALGORITHMS; COMPUTATIONAL METHODS; LAGRANGE MULTIPLIERS; PROBLEM SOLVING;

NONLINEAR PROGRAMMING;

EID: 1542337156     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10107-002-0352-8     Document Type: Article

References (23)

1. A. Barvinok. Problems of distance geometry and convex properties of quadratic maps. Discrete Computational Geometry, 13:189-202, 1995.
2. S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM Journal on Optimization, 10:443-461, 2000.
3. P.M. Bentler. A lower bound method for the dimension-free measurement of internal consistency. Social Science and Research, 1:343-357, 1972.
4. S. Burer and R.D.C. Monteiro. A projected gradient algorithm for solving the Maxcut SDP relaxation. Optimization Methods and Software, 15:175-200, 2001.
5. S. Burer, R.D.C. Monteiro, and Y. Zhang. Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation. Technical Report TR99-27, Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA, December 1999. Revised May 2001. Computational Optimization and Applications, 22:49-79, 2002.
6. S. Burer, R.D.C. Monteiro, and Y. Zhang. Solving a class of semidefinite programs via nonlinear programming. Manuscript, Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA, September 1999. Revised May 2001. Mathematical Programming, 93: 97-122, 2002.
7. C. Choi and Y. Ye. Solving sparse semidefinite programs using the dual scaling algorithm with an iterative solver. Manuscript, Department of Management Sciences, University of Iowa, Iowa City, IA 52242, USA, 2000.
8. Seventh DIMACS implementation challenge on semidefinite and related optimization problems, November 2000. See the website: http://dimacs.rutgers. edu/Challenges/Seventh/Instances/.
9. W. E. Donath and A. J. Huffman. Algorithms for partitioning graphs and computer logic based on eigenvectors of connection matrices. IBM Technical Disclosure Bulletin, 15, 1972.
10. W. E. Donath and A. J. Huffman. Lower bounds for the partitioning of graphs. IBM J. Res. and Devel., 17, 1973.
11. R. Fletcher. Practical Methods of Optimization. John Wiley & Sons, New York, second edition, 1987.
12. M. Fukuda and M. Kojima. Interior-point methods for lagrangian duals of semidefinite programs. Manuscript, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguru-ku, Tokyo 152, Japan, December 2000.
13. M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of ACM, pages 1115-1145, 42 (1995).
14. C. Helmberg and F. Rendl. A spectral bundle method for semidefinite programming. SIAM Journal on Optimization, 10:673-696, 2000.
15. S. Homer and M. Peinado. Design and performance of parallel and distributed approximation algorithms for maxcut. Journal of Parallel and Distributed Computing, 46:48-61, 1997.
16. L. Lovász. On the Shannon Capacity of a graph. IEEE Transactions of Information Theory, IT-25(1):1-7, January 1979.
17. H.-I Lu and P.N. Klein. Space-efficient approximation algorithms for MAXCUT and COLORING semidefinite programs. In Proceedings of the 8th Annual International Symposium on Algorithms and Computation (ISAAC'98), Taejon. Korea, Dec. 14-16, LNCS 1533, pages 387-396. Springer-Verlag, 1998.
18. R.D.C. Monteiro and M. Todd. Path-following methods for semidefinite programming. In R. Saigal, L. Vandenberghe, and H. Wolkowicz, editors, Handbook of Semidefinite Programming. Kluwer Academic Publishers, 2000.
19. G. Pataki. Cone-LP's and semidefinite programs: Geometry and a simplex-type method. In Proceedings of the Fifth IPCO Conference on Integer Programming and Combinatorial Optimization. Springer-Verlag, 1996.
20. G. Pataki. On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Mathematics of Operations Research, 23:339-358, 1998.
21. S. Poljak and F. Rendl. Nonpolyhendral relaxations of graph-bisection problems. SIAM Journal on Optimization, 5:467-487, 1995.
22. A. Shapiro. Minimum trace factor analysis. In S. Kotz and N.L. Johnson, editors, Encyclopedia of Statistical Sciences, volume 5. John Wiley and Sons, New York, 1985.
23. L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38:49-95, 1996.