Semidefinite optimization

Acta Numerica

Volumn 10, Issue , 2001, Pages 515-560

  Todd, M J ^a  

^a Department of Population Medicine and Diagnostic Sciences   (United States)

Author keywords

[No Author keywords available]

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EID: 0000920501     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492901000071     Document Type: Article

References (70)

1. F. Alizadeh (1995), Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5, 13-51.
2. F. Alizadeh (2001), Semidefinite programming home page: http://karush.rutgers.edu/~alizadeh/Sdppage/index.html
3. F. Alizadeh and S. Schmieta (2000), Symmetric cones, potential reduction methods and word-by-word extensions, in Handbook of Semidefinite Programming (H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds), Kluwer Academic Publishers, Boston/Dordrecht/London, pp. 195-233.
4. F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton (1994), A new primal-dual interior-point method for semidefinite programming, in Proceedings of the 5th SIAM Conference on Applied Linear Algebra (J. G. Lewis, ed.), SIAM, Philadelphia, USA, pp. 113-117.
5. F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton (1998), Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results, SIAM J. Optim. 8, 746-768.
6. R. Bellman and K. Fan (1963), On systems of linear inequalities in Hermitian matrix variables, in Convexity, Vol. 7 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, pp. 1-11.
7. A. Ben-Tal and A. Nemirovski (1998), Robust convex optimization, Math. Oper. Res. 23, 769-805.
8. S. J. Benson, Y. Ye, and X. Zhang (2000), Solving large-scale sparse semidefinite programs for combinatorial optimization, SIAM J. Optim. 10, 443-461.
9. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan (1994), Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, USA.
10. S. Burer, R. D. C. Monteiro, and Y. Zhang (1999a), Solving semidefinite programs via nonlinear programming II: Interior point methods for a subclass of SDPs, Technical Report TR99-23, Department of Computational and Applied Mathematics, Rice University, Houston, TX.
11. S. Burer, R. D. C. Monteiro, and Y. Zhang (1999b), Interior point algorithms for semidefinite programming based on a nonlinear programming formulation, Technical Report TR99-27, Department of Computational and Applied Mathematics, Rice University, Houston, TX.
12. J. Cullum, W. E. Donath, and P. Wolfe (1975), The minimization of certain nondif-ferentiable sums of eigenvalues of symmetric matrices, Math. Programming Study 3, 35-55.
13. W. E. Donath and A. J. Hoffman (1973), Lower bounds for the partitioning of graphs, IBM J. Research and Development 17, 420-425.
14. C. Delorme and S. Poljak (1993), Laplacian eigenvalues and the maximum cut problem, Math. Programming 62, 557-574.
15. L. Faybusovich (1997a), Euclidean Jordan algebras and interior-point algorithms, Positivity 1, 331-357.
16. L. Faybusovich (1997b), Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math. 86, 149-175.
17. R. Fletcher (1981), A nonlinear programming problem in statistics (educational testing), SIAM J. Sci. Statist. Comput. 2, 257-267.
18. R. Fletcher (1985), Semi-definite matrix constraints in optimization, SIAM J. Control Optim. 23, 493-513.
19. M. Fukuda and M. Kojima (2000), Interior-point methods for Lagrangian duals of semidefinite programs, Technical Report B-365, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.
20. M. Fukuda, M. Kojima, K. Murota, and K. Nakata (2000), Exploiting sparsity in semidefinite programming via matrix completions I: General framework, SIAM J. Optim. 11, 647-674.
21. M. X. Goemans (1997), Semidefinite programming in combinatorial optimization, Math. Programming 79, 143-161.
22. M. X. Goemans and D. P. Williamson (1995), Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42, 1115-1145.
23. D. Goldfarb and K. Scheinberg (1998), Interior point trajectories in semidefinite programming, SIAM J. Optim. 8, 871-886.
24. M. Grötschel, L. Lovász, and A. Schrijver (1988), Geometric Algorithms and Combinatorial Optimization, Springer, Berlin.
25. C. Helmberg (2001), Semidefinite programming home page: http://www.zib.de/helmberg/semidef.html
26. C. Helmberg and F. Oustry (2000), Bundle methods to minimize the maximum eigenvalue function, in Handbook of Semidefinite Programming (H. Wolkow-icz, R. Saigal, and L. Vandenberghe, eds), Kluwer Academic Publishers, Boston/Dordrecht/London, pp. 307-337.
27. C. Helmberg and F. Rendl (2000), A spectral bundle method for semidefinite programming, SIAM J. Optim. 10, 673-696.
28. C. Helmberg, F. Rendl, R. Vanderbei, and H. Wolkowicz (1996), An interior-point method for semidefinite programming, SIAM J. Optim. 6, 342-361.
29. F. Jarre (1999), A QQP-minimization method for semidefinite and smooth noncon-vex programs, Technical Report, Institut für Angewandte Mathematik und Statistik, Universität Würzburg, Germany.
30. N. K. Karmarkar (1984), A new polynomial-time algorithm for linear programming, Combinatorica 4, 373-395.
31. E. de Klerk, C. Roos, and T. Terlaky (1997), Initialization in semidefinite programming via a self-dual skew-symmetric embedding, Operations Research Letters 20, 213-221.
32. M. Kojima, S. Shindoh, and S. Hara (1997), Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM J. Optim. 7, 86-125.
33. M. Kojima, M. Shida, and S. Shindoh (1999), Search directions in the SDP and the monotone SDLCP: Generalization and inexact computation, Math. Programming 85, 51-80.
34. A. S. Lewis and M. L. Overton (1996), Eigenvalue optimization, in Acta Numerica, Vol. 5, Cambridge University Press, pp. 149-190.
35. L. Lovász (1979), On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25, 1-7.
36. L. Lovász and A. Schrijver (1991), Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim. 1, 166-190.
37. Z.-Q. Luo, J. F. Sturm, and S. Zhang (1998), Superlinear convergence of a symmetric primal-dual path-following algorithm for semidefinite programming, SIAM J. Optim. 8, 59-81.
38. Z.-Q. Luo, J. F. Sturm, and S. Zhang (2000), Conic convex programming and self-dual embedding, Optim. Methods Software 14, 169-218.
39. R. D. C. Monteiro (1997), Primal-dual path-following algorithms for semidefinite programming, SIAM J. Optim. 7, 663-678.
40. R. D. C. Monteiro (1998), Polynomial convergence of primal-dual algorithms for semidefinite programming based on the Monteiro and Zhang family of directions, SIAM J. Optim. 8, 797-812.
41. R. D. C. Monteiro and M. J. Todd (2000), Path-following methods, in Handbook of Semidefinite Programming (H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds), Kluwer Academic Publishers, Boston/Dordrecht/London, pp. 267-306.
42. R. D. C. Monteiro and P. R. Zanjacomo (1997), A note on the existence of the Alizadeh-Haeberly-Overton direction for semidefinite programming, Math. Programming, 78, 393-396.
43. R. D. C. Monteiro and Y. Zhang (1998), A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming, Math. Programming 81, 281-299.
44. Yu. E. Nesterov and A. S. Nemirovski (1992), Conic formulation of a convex programming problem and duality, Optim. Methods Software 1, 95-115.
45. Yu. E. Nesterov and A. S. Nemirovski (1994), Interior Point Polynomial Algorithms in Convex Programming, SIAM Publications, SIAM, Philadelphia, USA.
46. Yu. E. Nesterov and M. J. Todd (1997), Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res. 22, 1-42.
47. Yu. E. Nesterov and M. J. Todd (1998), Primal-dual interior-point methods for self-scaled cones, SIAM J. Optim. 8, 324-364.
48. F. Oustry (1999), The U-Lagrangian of the maximum eigenvalue function, SIAM J. Optim. 9, 526-549.
49. F. Oustry (2000), A second-order bundle method to minimize the maximum eigenvalue function, Math. Programming 89, 1-33.
50. M. L. Overton and R. S. Womersley (1993), Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Math. Programming 62, 321-357.
51. S. Poljak and Z. Tuza (1995), Maximum cuts and largest bipartite subgraphs, in Combinatorial Optimization (W. Cook, L. Lovász, and P. Seymour, eds), American Mathematical Society, Providence, RI, pp. 181-244.
52. S. Poljak, F. Rendl, and H. Wolkowicz (1995), A recipe for semidefinite relaxation for (0, 1)-quadratic programming, J. Global Optimization 7, 51-73.
53. F. Potra and R. Sheng (1998), On homogeneous interior-point algorithms for semi-definite programming, Optim. Methods Software 9, 161-184.
54. M. Ramana (1997), An exact duality theory for semidefinite programming and its complexity implications, Math. Programming 77, 129-162.
55. M. Ramana, L. Tunçel, and H. Wolkowicz (1997), Strong duality for semidefinite programming, SIAM J. Optim. 7, 641-662.
56. R. T. Rockafellar (1970), Convex Analysis, Princeton University Press, Princeton, NJ.
57. N. Z. Shor (1990), Dual quadratic estimates in polynomial and Boolean programming, Ann. Oper. Res. 25, 163-168.
58. K. Tanabe (1988), Centered Newton method for mathematical programming, in System Modeling and Optimization, Springer, NY, pp. 197-206.
59. M. J. Todd (1999), A study of search directions in interior-point methods for semi-definite programming, Optim. Methods Software 11&12, 1-46.
60. M. J. Todd and Y. Ye (1990), A centered projective algorithm for linear programming, Math. Oper. Res. 15, 508-529.
61. M. J. Todd, K.-C. Toh, and R. H. Tütüncü (1998), On the Nesterov-Todd direction in semidefinite programming, SIAM J. Optim. 8, 769-796.
62. L. Tunçel (2000), Potential reduction and primal-dual methods, in Handbook of Semidefinite Programming (H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds), Kluwer Academic Publishers, Boston/Dordrecht/London, pp. 235-265.
63. L. Vandenberghe and S. Boyd (1996), Semidefinite programming, SIAM Rev. 38, 49-95.
64. R. J. Vanderbei and H. Yurttan Benson (1999), On formulating semidefinite programming problems as smooth convex nonlinear optimization problems, Report ORFE 99-01, Operations Research and Financial Engineering, Princeton, NJ.
65. S. A. Vavasis (1999), A note on efficient computation of the gradient in semidefinite programming, Technical Report, Department of Computer Science, Cornell University, Ithaca, NY.
66. H. Wolkowicz (2001), Bibliography on semidefinite programming: http:liinwww.ira.uka.de/bibliography/Math/psd.html
67. H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds (2000), Handbook of Semidefin-ite Programming, Kluwer Academic Publishers, Boston/Dordrecht/London.
68. S. J. Wright (2001), Interior-point methods online home page: http://www-unix.mcs.anl.gov/otc/InteriorPoint/
69. Y. Ye, M. J. Todd, and S. Mizuno (1994), An O(√nL)-iteration homogeneous and self-dual linear programming algorithm, Math. Oper. Res. 19, 53-67.
70. Y. Zhang (1998), On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM J. Optim. 8, 365-386.