Semidefinite relaxations for integer programming

50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art

Volumn , Issue , 2010, Pages 687-726

  Rendl, Franz ^a  

^a UNIVERSITY OF KLAGENFURT   (Austria)

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[No Author keywords available]

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EID: 84885784676     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-68279-0_18     Document Type: Chapter

References (74)

1. K.M. Anstreicher and H. Wolkowicz, On Lagrangian relaxation of quadratic matrix constraints, SIAM Journal on Matrix Analysis 22 (2000) 41-55.
2. S. Arora, E. Chlamtac, and M. Charikar, New approximation guarantee for chromatic number, Proceedings of the 38th STOC, Seattle, USA, 2006, pp. 215-224.
3. D. Avis and J. Umemoto, Stronger linear programming relaxations for max-cut, Mathematical Programming 97 (2003) 451-469.
4. E. Balas, S. Ceria, and G. Cornuéjols, A lift-and-project cutting plane algorithm for mixed 0-1 programs, Mathematical Programming 58 (1993) 295-324.
5. F. Barahona, M. Jünger, and G. Reinelt, Experiments in quadratic 0-1 programming, Mathematical Programming 44 (1989) 127-137.
6. A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization, MPS-SIAM Series on Optimization, 2001.
7. S. Berkowitz, Extrema of elementary symmetric polynomials of the eigenvalues of the matrix PKP +L, Linear Algebra Appl. 8 (1974) 273-280.
8. S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming 120 (2009) 479-495.
9. E. Chlamtac, Non-local analysis of sdp based approximation algorithms, Ph.D. thesis, Princeton University, USA, 2009.
10. G.B. Dantzig, D.R. Fulkerson, and S.M. Johnson, Solution of a large scale traveling salesman problem, Journal of the Operations Research Society of America 2 (1954) 393-410.
11. E. de Klerk and D.V. Pasechnik, Approximatin of the stability number of a graph via copositive programming, SIAM Journal on Optimization 12 (2002) 875-892.
12. M. Deza, V.P. Grishukhin, and M. Laurent, The hypermetric cone is polyhedral, Combinator-ica 13 (1993) 397-411.
13. W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs, IBM Journal of Research and Developement 17 (1973) 420-425.
14. R.J. Duffin, Infinite programs, Ann. Math. Stud. 38 (1956) 157-170.
15. I. Dukanovic and F. Rendl, Semidefinite programming relaxations for graph coloring and maximal clique problems, Mathematical Programming 109 (2007) 345-365.
16. D.V. Pasechnik, E. de Klerk, and J.P. Warners, On approximate graph colouring and MAX-k-CUT algorithms based on the v-function, Journal of Combinatorial Optimization 8 (2004) 267-294.
17. I. Fischer, G. Gruber, F. Rendl, and R. Sotirov, Computational experience with a bundle method for semidefinite cutten plane relaxations of max-cut and equipartition, Mathematical Programming 105 (2006) 451-469.
18. A. Frieze and M. Jerrum, Improved approximation algorithms for MAX k-cut and MAX BISECTION, Algorithmica 18 (1997) 67-81.
19. M.X. Goemans, Semidefinite programming in combinatorial optimization, Mathematical Programming 79 (1997) 143-162.
20. M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM 42 (1995) 1115-1145.
21. N. Gvozdenović and M. Laurent, Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization, SIAM Journal on Optimization 19 (2008) 592-615.
22. N. Gvozdenović and M. Laurent, The operatorΨ for the chromatic number of a graph, SIAM Journal on Optimization 19 (2008) 572-591.
23. S.W. Hadley, F. Rendl, and H. Wolkowicz, A new lower bound via projection for the quadratic assignment problem, Mathematics of Operations Research 17 (1992) 727-739.
24. E. Halperin and U. Zwick, A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems, Lecture notes in Computer Science 2081, IPCO 2001, Springer Berlin, 2001, pp. 210-225.
25. P.L. Hammer, Some network flow problems solved with pseudo-Boolean programming, Operations Research 13 (1965) 388-399.
26. C. Helmberg, Fixing variables in semidefinite relaxations, SIAM J. Matrix Anal. Appl. 21 (2000) 952-969.
27. C. Helmberg, Semidefinite programming, European Journal of Operational Research 137 (2002) 461-482.
28. C. Helmberg, Numerical validation of SBmethod, Mathematical Programming 95 (2003) 381-406.
29. C. Helmberg, K.C. Kiwiel, and F. Rendl, Incorporating inequality constraints in the spectral bundle method, Integer Programming and combinatorial optimization (E.A. Boyd R.E. Bixby and R.Z. Rios-Mercado, eds.), Springer Lecture Notes in Computer Science 1412, 1998, pp. 423-435.
30. C. Helmberg and F. Oustry, Bundle methods to minimize the maximum eigenvalue function, Handbook of semidefinite programming: theory, algorithms and applications (R. Saigal H. Wolkowicz and L. Vandenberghe, eds.), Kluwer, 2000, pp. 307-337.
31. C. Helmberg, S. Poljak, F. Rendl, and H. Wolkowicz, Combining semidefinite and polyhedral relaxations for integer programs, Integer Programming and combinatorial optimization (E. Balas and J. Clausen, eds.), Springer Lecture Notes in Computer Science 920, 1995, pp. 124-134.
32. C. Helmberg and F. Rendl, Solving quadratic (0,1)-problems by semidefinite programming and cutting planes, Mathematical Programming 82 (1998) 291-315.
33. C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM Journal on Optimization 10 (2000) 673-696.
34. C. Helmberg, F. Rendl, R. Vanderbei, and H. Wolkowicz, An interior-point method for semidefinite programming, SIAM Journal on Optimization 6 (1996) 342-361.
35. J.B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms (Vol. 1 and 2), Springer, 1993.
36. A.J. Hoffman and H.W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. Journal 20 (1953) 37-39.
37. D. Jibetean and M. Laurent, Semidefinite approximations for global unconstrained polynomial optimization, SIAM Journal on Optimization 16 (2005) 490-514.
38. D.S. Johnson, C.R. Aragon, L.A. McGeoch, and C. Schevon, Optimization by simulated annealing: An experimental evaluation. I: Graph partitioning, Operations Research 37 (1989) 865-892.
39. D. Karger, R. Motwani, and M. Sudan, Approximate graph colouring by semidefinite programming, Journal of the ACM 45 (1998) 246-265.
40. S.E. Karisch and F. Rendl, Semidefinite programming and graph equipartition, Fields Institute Communications 18 (1998) 77-95.
41. B.W. Kernighan and S. Lin, An efficient heuristic procedure for partitioning graphs, Bell System techn. Journal 49 (1970) 291-307.
42. S. Khanna, N. Linial, and S. Safra, On the hardness of approximating the chromatic number, Combinatorica 20 (2000) 393-415.
43. M. Kojima, S. Shindoh, and S. Hara, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM Journal on Optimization 7 (1997) 86-125.
44. J.B. Lasserre, A sum of squares approximation of nonnegative polynomials, SIAM Journal Optimization 16 (2006) 751-765.
45. M. Laurent, S. Poljak, and F. Rendl, Connections between semidefinite relaxations of the maxcut and stable set problems, Mathematical Programming 77 (1997) 225-246.
46. M. Laurent and F. Rendl, Semidefinite programming and integer programming, Discrete Optimization (K. Aardal, G.L. Nemhauser, and R. Weismantel, eds.), Elsevier, 2005, pp. 393-514.
47. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys (eds.), The traveling salesman problem, a guided tour of combinatorial optimization, Wiley, Chicester, 1985.
48. C. Lemaréchal, An extension of davidon methods to nondifferentiable problems, Mathematical Programming Study 3 (1975) 95-109.
49. C. Lemaréchal, Nonsmooth optimization and descent methods, Tech. report, International Institute for Applied Systems Analysis, 1978.
50. A. Lisser and F. Rendl, Graph partitioning using linear and semidefinite programming, Mathematical Programming 95 (2002) 91-101.
51. L. Lovász, On the shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979) 1-7.
52. L. Lovász, Semidefinite programs and combinatorial optimization, Recent advances in algorithms and combinatorics (B.A. Reed and C.L. Sales, eds.), CMS books in Mathematics, Springer, 2003, pp. 137-194.
53. L. Lovász and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM Journal on Optimization 1 (1991) 166-190.
54. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, Proceedings of the 25th ACM STOC, 1993, pp. 286-293.
55. M. Marcus, Rearrangement and extremal results for Hermitian matrices, Linear Algebra Appl. 11 (1975) 95-104.
56. R.J. McEliece, E.R. Rodemich, and H.C. Rumsey Jr., The lovasz bound and some generalizations, Journal of combinatorics and System Sciences 3 (1978) 134-152.
57. R.D.C. Monteiro, Primal-dual path-following algorithms for semidefinite programming, SIAM Journal on Optmization 7 (1997) 663-678.
58. T.S. Motzkin and E.G. Straus, Maxima for graphs and a new proof of a theorem of turan, Canadian Journal of Mathematics 17 (1965) 533-540.
59. K.G. Murty and S.N. Kabadi, Some np-complete problems in quadratic and nonlinear programming, Mathematical Programming 39 (1987) 117-129.
60. Y. Nesterov, Quality of semidefinite relaxation for nonconvex quadratic optimization, Tech. report, CORE, 1997.
61. Y. Nesterov and A.S. Nemirovski, Interior point polynomial algorithms in convex programming, SIAM Publications, SIAM, Philadelphia, USA, 1994.
62. M. Padberg, The quadric Boolean polytope: some characteristics, facets and relatives, Mathematical Programming 45 (1989) 139-172.
63. P. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph.D. thesis, California Institute of Technology, USA, 2000.
64. J. Povh and F. Rendl, Approximating non-convex quadratic programs by semidefinite and copositive programming, Proceedings of the 11th international conference on operational research (L. Neralic V. Boljuncic and K. Soric, eds.), Croation Operations Research Society, 2008, pp. 35-45.
65. F. Rendl and H. Wolkowicz, Applications of parametric programming and eigenvalue maximization to the quadratic assignment problem, Mathematical Programming 53 (1992) 63-78.
66. F. Rendl and H. Wolkowicz, A projection technique for partitioning the nodes of a graph, Annals of Operations Research 58 (1995) 155-179.
67. H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results, SIAM Journal Optimization 2 (1992) 121-152.
68. A. Schrijver, A comparison of the delsarte and lovasz bounds, IEEE Transactions on Information Theory IT-25 (1979) 425-429.
69. H.D. Sherali and W.P. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems, SIAM Journal on Discrete Mathematics 3 (1990) 411-430.
70. H.D. Sherali and W.P. Adams, A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems, Discrete Applied Mathematics 52 (1994) 83-106.
71. C. De Simone, The cut polytope and the Boolean quadric polytope, Discrete Mathematics 79 (1990) 71-75.
72. J. von Neumann, Some matrix inequalities and metrization of matrix space (1937), John von Neumann: Collected Works, Vol 4, MacMillan, 1962, pp. 205-219.
73. A. Widgerson, Improving the performance guarantee for approximate graph colouring, Journal of the ACM 30 (1983) 729-735.
74. H. Wolkowicz, R. Saigal, and L. Vandenberghe (eds.), Handbook of semidefinite programming, Kluwer, 2000.