Fixing variables in semidefinite relaxations

SIAM Journal on Matrix Analysis and Applications

Volumn 21, Issue 3, 2000, Pages 952-969

  Helmberg, Christoph ^a  

^a ZUSE INSTITUTE BERLIN   (Germany)

Author keywords

Quadratic 0 1 programming; Reduced cost fixing; Semidefinite programming; Semidefinite relaxations

Indexed keywords

EID: 0034342222     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S089547989631442X     Document Type: Article

References (36)

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. A. HAEBERLY, AND M. L. OVERTON, Complementarity and nondegeneracy in semidefinite programming, Math. Programming, 77 (1997), pp. 111-128.
3. E. BALAS, S. CERIA, AND G. CORNUEJOLS, A lift-and-project cutting plane algorithm for mixed 0/1 programs, Math. Programming, 58 (1993), pp. 295-324.
4. F. BARAHONA, M. JÜNGER, AND G. REINELT, Experiments in quadratic 0-1 programming, Math. Programming, 44 (1989), pp. 127-137.
5. S. BENSON, Y. YE, AND X. ZHANG, Solving large-scale sparse semidefinite programs for combinatorial optimization, SIAM J. Optim., 10 (2000), pp. 443-461.
6. B. CHOR AND M. SUDAN, A geometric approach to betweenness, in Proceedings ESA'95, Lecture Notes in Comput. Sci. 979, Springer, New York, 1995, pp. 227-237.
7. C. DE SIMONE, The cut polytope and the boolean quadric polytope, Discrete Math., 79 (1989), pp. 71-75.
8. C. DELORME AND S. POLJAK, Laplacian eigenvalues and the maximum cut problem, Math. Programming, 62 (1993), pp. 557-574.
9. U. FEIGE AND M. X. GOEMANS, Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT, in Proceedings 3rd Israel Symposium on Theory of Computing and Systems, Tel Aviv, Israel, 1995, pp. 182-189.
10. A. FRIEZE AND M. JERRUM, Improved approximation algorithms for MAX k-CUT and MAX BISECTION, in Integer Programming and Combinatorial Optimization, E. Balas and J. Clausen, eds., Lecture Notes in Comput. Sci. 920, Springer, New York, 1995, pp. 1-13.
11. M. X. GOEMANS AND D. P. WILLIAMSON, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.
12. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
13. M. GROTSCHEL, L. LOVÁSZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, 2nd ed., Algorithms and Combinatorics 2, Springer, New York, 1988.
14. P. L. HAMMER, Some network flow problems solved with pseudo-boolean programming, Oper. Res., 13 (1965), pp. 388-399.
15. C. HELMBERG, Fixing variables in semidefinite relaxations, in Algorithms - ESA'97, R. Burkard and G. Woeginger, eds., Lecture Notes in Comput. Sci. 1284, Springer, New York, 1997, pp. 259-270.
16. C. HELMBERG, S. POLJAK, F. RENDL, AND H. WOLKOWICZ, Combining semidefinite and polyhedral relaxations for integer programs, in Integer Programming and Combinatorial Optimization, E. Balas and J. Clausen, eds., Lecture Notes in Comput. Sci. 920, Springer, New York, 1995, pp. 124-134.
17. C. HELMBERG AND F. RENDL, A spectral bundle method for semidefinite programming, SIAM J. Optim., 10 (2000), pp. 673-698.
18. C. HELMBERG AND F. RENDL, Solving quadratic (0, 1)-problems by semidefinite programs and cutting planes, Math. Programming, 82 (1998), pp. 291-315.
19. 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.
20. C. HELMBERG, F. RENDL, AND R. WEISMANTEL, Quadratic knapsack relaxations using cutting planes and semidefinite programming, in Integer Programming and Combinatorial Optimization, W. H. Cunningham, S. T. McCormick, and M. Queyranne, eds., Lecture Notes in Comput. Sci. 1084, Springer, New York, 1996, pp. 175-189.
21. 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.
22. D. KARGER, R. MOTWANI, AND M. SUDAN, Approximate graph coloring by semidefinite programming, J. ACM, 45 (1998), pp. 246-265
23. S. E. KARISCH AND F. RENDL, Semidefinite programming and graph equipartition, in Topics in Semidefinite and Interior-Point Methods, P.M. Pardalos and H. Wolkowicz, eds., Field Inst. Commun. 18, AMS, Providence, RI, 1998, pp. 77-95.
24. M. KOJIMA, S. SHINDOH, AND S. HARA, Interior-Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices, SIAM J. Optim., 7 (1997), pp. 86-125
25. M. LAURENT, S. POLJAK, AND F. RENDL, Connections between semidefinite relaxations of the max-cut and stable set problems, Math. Programming, 77 (1997), pp. 225-246.
26. R. B. LEHOUCQ, D. C. SORENSEN, AND C. YANG, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods., Department of Computation and Applied Mathematics, Rice University, Houston, TX 1997, manuscript.
27. L. LOVÁSZ, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), pp. 1-7.
28. L. LOVASZ AND A. SCHRIJVER, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1 (1991), pp. 166-190.
29. Y. NESTEROV AND A. NEMIROVSKII, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics 13, SIAM, Philadelphia, PA, 1994.
30. Y. NESTEROV AND M. J. TODD, Self-scaled barriers and interior-Point methods for convex programming, Math. Oper. Res., 22 (1997), pp. 1-42.
31. S. POLJAK AND F. RENDL, Nonpolyhedral relaxations of graph-bisection problems, SIAM J. Optim., 5 (1995), pp. 467-487.
32. L. TUNÇEL, Primal-dual symmetry and scale invariance of interior-point algorithms for convex optimization, Math. Oper. Res., 23 (1998), pp. 708-718.
33. L. VANDENBERGHE AND S. BOYD, A primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, Ser. B, 69 (1995), pp. 205-236.
34. A. WILLIAMS, Quadratic 0-1 Programming Using the Roof Dual with Computational Results, RUTCOR Research Report 8-85, Rutgers University, Piscataway, NJ, 1985.
35. H. WOLKOWICZ AND Q. ZHAO, Semidefinite Programming Relaxations for the Graph Partitioning Problem, J. Combin. Optim., 2 (1998), pp. 71-109.
36. Q. ZHAO, S. E. KARISCH, F. RENDL, AND H. WOLKOWICZ, Semidefinite Programming Relaxations for the Quadratic Assignment Problem, CORR Report 95/27, University of Waterloo, Ontario, Canada, 1996.