From linear to semidefinite programming: An algorithm to obtain semidefinite relaxations for bivalent quadratic problems

Journal of Combinatorial Optimization

Volumn 8, Issue 4, 2004, Pages 469-493

  Roupin, Frédéric ^a  

^a CEDRIC   (France)

Author keywords

Constrained memory allocation problem; K CLUSTER problem; Quadratic assignment problem; Semidefinite programming

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMBINATORIAL MATHEMATICS; CONSTRAINT THEORY; MATRIX ALGEBRA; OPTIMIZATION; PROBLEM SOLVING; QUADRATIC PROGRAMMING; RELAXATION PROCESSES; SET THEORY; STORAGE ALLOCATION (COMPUTER);

CONSTRAINED-MEMORY ALLOCATION PROBLEM; K-CLUSTER PROBLEM; QUADRATIC ASSIGNMENT PROBLEM; SEMIDEFINITE PROGRAMMING;

LINEAR PROGRAMMING;

EID: 14844362280     PISSN: 13826905     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10878-004-4838-6     Document Type: Article

References (42)

1. W.P. Adams and H.D. Sherali, "A tight linearization and an algorithm for zero-one quadratic programming problems," Management Science, vol. 32, no. 10, pp. 1274-1290, 1986.
2. W.P. Adams and H.D. Sherali, "A hierarchy of relaxations between the continuous and convex hull representations pf zero-one programming problems," SIAM J. Disc. Math, vol. 3, pp. 411-430, 1990.
3. F. Alizadeh, "Interior point methods in semidefinite programming with applications to combinatorial optimization," SIAM J. Optim., vol. 5, no. 1, pp. 13-51, 1995.
4. M.F. Anjos, "An improved semidefinite programming relaxation for the satisfiability problem," Technical Report UW-E&CE#2002-09, Department of Electrical & Computer Engineering, University of Waterloo, Canada, June 2002.
5. K.M. Anstreicher and N. Brixius, "A new bound for the quadratic assignment problem based on convex quadratic programming," Mathematical Programming, vol. 89, pp. 341-357, 2001.
6. K.M. Anstreicher, N. Brixius, J.-P. Goux, and J. Linderoth, "Solving large quadratic assignment problems on computational grids," Mathematical Programming B, vol. 91, pp. 563-588, 2002.
7. E. Balas, S. Ceria, and G. Cornuejols, "A lift-and-project cutting plane algorithm for mixed 0-1 programs," Math. Prog. A, vol. 58, pp. 295-323, 1993.
8. S. Benson, Y. Ye, and X. Zhang, "Solving large-scale sparse semidefinite programs for combinatorial optimization," SIAM J. Optim., vol. 10, no. 2, pp. 443-161, 2000.
9. A. Billionnet, "Different formulations for solving the heaviest k-subgraph problem," Rapport de recherche CEDRIC 384-2002 (http://cedric.cnam.fr), 2002.
10. S. Burer and R.D. Monteiro, "A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization," Technical Report, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332. To appear in Math. Programming, Series B.
11. R.E. Burkard, S. Karisch, and F. Rendl, "A quadratic assignment problem library," Journal of Global Optimization, vol. 10, pp. 391-403, 1997.
12. G. Delaporte, S. Jouteau, and F. Roupin, "A Tool to formulate and solve semidefinite relaxations for bivalent quadratic problems," in Proceedings ROADEF 2003, Avignon 26-28 Février, 2003. http://semidef.free.fr.
13. S. Elloumi, "Contributions à l'optimisation combinatoire," Habilitation à diriger des recherches, Université de Paris Nord, France, 26 Juin 2002.(http://cedric.cnam.fr).
14. R. Fortet, "L'algèbre de boole et ses applications en recherche opérationnelle," Cahier du centre d'étude de Recherche Opérationnelle, vol. 4, pp. 5-36, 1959.
15. M.X. Goemans and S.P. Williamson, "Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming," Journal of the ACM, vol. 42, no. 6, pp. 1115-1145, 1995.
16. Q. Han, Y. Ye, and J. Zhang, "An improved rounded method and semidefinite programming relaxation for graph partition," Mathematical Programming, vol. 92, no. 3, pp. 509-535, 2002.
17. Handbook of Semidefinite Programming, Theory, Algorithms, and Applications, H. Wolkowicz, R. Saigal, and L. Vandenberghe (Eds.), Kluwer Academic Publishers, 2000.
18. C. Hemberg, "A C++ implementation of the spectral bundle method," Manual version 1.1.1 http://www.zib.de/helmberg/method.
19. C. Helmberg and F. Rendl, "Solving quadratic (0,1)-problems by semidefinite programs and cutting planes," Math. Programming, Serie A, vol. 82, no. 3, pp. 291-315, 1998.
20. C. Helmberg and K.C. Kiwiel, "A spectral bundle method with bounds," ZIB Preprint SC-99-37, Konrad-Zuse-Zentrum Berlin, December 1999.
21. C. Helmberg, F. Rendl, and R. Weismantel, "A semidefinite programming approach to the quadratic knapsack problem," J. of Combinatorial Optimization, vol. 4, pp. 197-215, 2000.
22. C. Helmberg, "Semidefinite programming for combinatorial optimization," Habilitationsschrift TU Berlin, ZIB-report ZR-00-34, Konrad-Zuse-Zentrum für Informationstechnick Berlin, Takustrasee 7, 14195 Berlin, Germany, 2000.
23. C. Helmberg and F. Rendl, "A spectral bundle method for semidefinite programming," SIAM J. Optim., vol. 10, no. 3, pp. 673-696, 2000.
24. C. Helmberg, "Cutting planes algorithm for large scale semidefinite relaxations," ZIB-Report ZR 01-26, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustraße 7, 14195 Berlin, Germany, Oct. 2001.
25. C. Helmberg, "Numerical evaluation of SBmethod," ZIB-Report ZR 01-37, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustraße 7, 14195 Berlin, Germany, 2001.
26. G. Iyengar and M.T. Cezik, "Cutting planes for mixed 0-1 mixed semidefinite programs," in Integer and Combinatorial Optimization, vol. 2081 of Lectures Notes in Computer Science, K. Aardal and B. Gerards (Eds.), Springer, 2001, pp. 251-263.
27. G. Iyengar, "Quadratic cuts for mixed 0-1 quadratic programs," Technical Report, IEOR Dept., Columbia University, 2001.
28. J.B. Lasserre, "An explicit exact SDP relaxation for nonlinear 0-1 programs," SIAM J. Optim., vol. 12, pp. 756-769, 2002.
29. J.B. Lasserre, "Semidefinite programming vs. LP relaxations for polynomial programming," Math. Oper. Res., pp. 347-360, 2002.
30. M. Laurent, S. Poljak, and F. Rendl, "Connections between semidefinite relaxations of the max-cut and stable set problems," Mathematical Programming, vol. 77, pp. 225-246, 1997.
31. M. Laurent, "A comparison of the Sherali-Adams, Lovász- Schrijver and Lasserre relaxations for 0-1 programming," Mathematics of Operations Research, vol. 28, pp. 470-496, 2003.
32. C. Lemarechal and F. Oustry, "Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization," RR-3710, INRIA Rhone-Alpes, ZIRST - 655 avenue de l'Europe, F-38330 Montbonnot Saint-Martin, June 1999.
33. L. Lovász, "On the Shannon capacity of a graph," IIIE Transactions on Information Theory, vol. IT-25, no. 1, pp. 1-7, 1979.
34. L. Lovász and A. Schrijver, "Cones of matrices and set functions and 0-1 optimization," SIAM J. on Optimization, vol. 1, pp. 166-190, 1991.
35. J.E. Mitchell and K. Krishnan, "Cutting plane methods for semidefinite programming," Technical Report, Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180 USA, Nov. 2002.
36. H.D. Mittelmann, "An independent benchmarking of SDP and SOCP solvers," Technical Report, Dept. Mathematics, Arizona State University, Tempe, AZ 85287-1804, 2001.
37. Y. Nesterov and A. Nemirovski, "Interior-point polynomial algorithms in convex programming," SIAM Studies in Applied Mathematics, Philadelphia, 1994.
38. S. Poljak, F. Rendl, and H. Wolkowicz, "A recipe for semidefinite relaxation for (0,1)-quadratic programming," Journal of Global Optimization, vol. 7, pp. 51-73, 1995.
39. M.G.C. Resende, K.G. Ramakrisnhnan, and Z. Drezner, "Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming," Operations Research, vol. 43, no. 5, pp. 781-791, 1995.
40. N.Z. Shor, "Quadratic optimization problems," Soviet Journal of computer Systems Sciences, vol. 25, pp. 1-11, 1987.
41. J.F. Sturm, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones," Optimization Methods and Software, vol. 11, pp. 625-653, 1999.
42. Q. Zhao, S.E. Karish, F. Rendl, and H. Wolkowicz, "Semidefinite programming relaxations for the quadratic assignment problem," J. of Combinatorial Optimization, vol. 2, no. 1, pp. 71-109, 1998.