The gauss-newton direction in semidefinite programming

Optimization Methods and Software

Volumn 15, Issue 1, 2001, Pages 1-28

  Kruk, Serge ^a   Muramatsu, Masakazu ^b   Rendl, Franz ^c   Vanderbei, Robert J ^d   Wolkowicz, Henry ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

^b SOPHIA UNIVERSITY   (Japan)

^c UNIVERSITY OF KLAGENFURT   (Austria)

^d PRINCETON UNIVERSITY   (United States)

Author keywords

Newton direction; Semidefinite programming; Symmetrization

Indexed keywords

ALGORITHMS; LINEAR SYSTEMS; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES;

SEMIDEFINITE PROGRAMMING (SDP);

LINEAR PROGRAMMING;

EID: 0036454456     PISSN: 10556788     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (69)

1. Alfakih, A., Khandani, A. and Wolkowicz, H. (1999). Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl., 12(1-3), 13-30. Computational optimization- a tribute to Olvi Mangasarian, Part I.
2. Alizadeh, F., Combinatorial optimization with interior point methods and semidefinite matrices. Ph.D. Thesis, University of Minnesota, 1991.
3. Alizadeh, F. (1995). Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim., 5, 13-51.
4. Alizadeh, F., Haeberly, J.-P., Nayakkankuppam, M. V., Overton, M. L. and Schmieta, S., SDPpack user's guide - version 0.9 Beta. Technical Report TR 1997-737, Courant Institute of Mathematical Sciences, NYU, New York, NY, June, 1997.
5. Alizadeh, F., Haeberly, J.-P. A. and Overton, M. L. (1997). Complementarity and nondegeneracy in semidefinite programming. Math. Programming, 77, 111-128.
6. Alizadeh, F., Haeberly, J.-P. A. and Overton, M. L. (1998). Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. SIAM J. Optim., 8, 746-768.
7. Ben-israel, A. and Greville, T. N. E., Generalized Inverses: Theory and Applications. Wiley-Interscience, 1974.
8. Adi Ben-israel, Newton's method with modified functions. In: Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995), Volume 204 of Contemp. Math., pp. 39-50. Amer. Math. Soc., Providence, RI, 1997.
9. Bjorck, A., Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996.
10. Boyd, S., Balakrishnan, V., Feron, E. and El Ghaoui, L. (1993). Control system analysis and synthesis via linear matrix inequalities. Proc. ACC, pp. 2147-2154.
11. Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Volume 15 of Studies in Applied Mathematics. SIAM, Philadelphia, PA, June, 1994.
12. Brixius, N., Potra, F. A. and Sheng, R., SDPHA: a matlab implementation of homogeneous interior-point algorithms for semidefinite programming. Technical Report, University of Iowa, Iowa City, IA, 1998.
13. Dennis, J. E. Jr. and Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Volume 16 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Corrected reprint of the 1983 original.
14. El-bakry, A. S., Tapia, R. A., Tsuchiya, T. and Zhang, Y. (1996). On the formulation and theory of the Newton interior-point method for nonlinear programming. J. Optim. Theory Applic., 89, 507-541.
15. Elhallabi, M., An inexact hybrid algorithm for nonlinear systems of equations. Technical Report CRPC-TR94372, Department of Mathematical Sciences, Rice University, Houston, TX, 1994.
16. Elhallabi, M. and Tapia, R. A., An inexact trust-region feasible-point algorithm: globalization of Gauss-Newton method. Technical Report TR93-43, Department of Mathematical Sciences, Rice University, Houston, TX, 1993.
17. Elhallabi, M. and Tapia, R. A., An inexact trust-region feasible-point algorithm for nonlinear systems of equalities and inequalities. Technical Report CRPC-TR95543, Department of Mathematical Sciences, Rice University, Houston, TX, 1995.
18. Fletcher, R., Practical Methods of Optimization, Vol. 1, Unconstrained Optimization. John Wiley & Sons, New York, NY, 1980.
19. Fujisawa, K., Kojima, M. and Nakata, K., SDPA semidefinite programming algorithm. Technical Report, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1995.
20. Gerlach, J. (1994). Accelerated convergence in Newton's method. SIAM Review, 36, 272-276.
21. Goemans, M. X. (1997). Semidefinite programming in combinatorial optimization. Math. Programming, 79, 143-162.
22. Gu, M., On primal-dual interior point methods for semidefinite programming. Technical Report CAM 97-12, Department of Mathematics, University of California, Los Angeles, CA 90095-1555, 1997.
23. Helmberg, C., An interior-point method for semidefinite programming and max-cut bounds. Ph.D. Thesis, Graz University of Technology, Austria, 1994.
24. Helmberg, C., Rendl, F., Vanderbei, R. J. and Wolkowicz, H. (1996). An interior-point method for semidefinite programming. SIAM J. Optim., 6(2), 342-361.
25. Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis. Cambridge University Press, New York, 1991.
26. Johnson, C. R., Kroschel, B. and Wolkowicz, H. (1998). An interior-point method for approximate positive semidefinite completions. Comput. Optim. Appl., 9(2), 175-190.
27. Karisch, S., Nonlinear Approaches for Quadratic Assignment and Graph Partition Problems. Ph.D. Thesis, University of Graz, Graz, Austria, 1995.
28. De Klerk, E., Interior point methods for semidefinite programming. Ph.D. Thesis, Delft University, 1997.
29. De Klerk, E., Peng, J., Roos, C. and Terlaky, T., A scaled Gauss-Newton primal-dual search direction for semidefinite programming. Technical Report, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Delft, The Netherlands, 1999.
30. Kojima, M., Megiddo, N. and Mizuno, S. (1993). A primal-dual infeasible-interior-point algorithm for linear programming. Math. Programming, 61, 263-280.
31. Kojima, M., Mizuno, S. and Yoshise, A., A primal-dual interior point algorithm for linear programming. In: Megiddo, N. Ed., Progress in Mathematical Programming: Interior Point and Related Methods, pp. 29-47. Springer Verlag, New York, 1989.
32. Kojima, M., Shindoh, S. and Hara, S. (1997). Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim., 7(1), 86-125.
33. Kruk, S., Semidefinite programming applied to nonlinear programming. Master's Thesis, University of Waterloo, 1996.
34. Kruk, S., Muramatsu, M., Rendl, F., Vanderbei, R. J. and Wolkowicz, H., The Gauss-Newton direction in linear and semidefinite programming. Technical Report CORR 98-16, University of Waterloo, Waterloo, Canada, 1998.
35. Megiddo, N., Pathways to the optimal set in linear programming. In: Megiddo, N. Ed., Progress In Mathematical Programming: Interior Point and Related Methods, pp. 131-158. Springer Verlag, New York, 1989.
36. Identical version In: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, pp. 1-35, 1986.
37. Monteiro, R. D. C. (1997). Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7(3), 663-678.
38. Monteiro, R. D. C. and Zanjácomo, P. R. (1997). A note on the existence of the Alizadeh-Haeberly-Overton direction for semidefinite programming. Math. Programming, 78(3, Ser. A), 393-396.
39. Monteiro, R. D. C. and Zhang, Y. (1998). A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming. Math. Programming, 81(3, Ser. A), 281-299.
40. Moré, J. J. and Sorensen, D. C. (1983). Computing a trust region step. SIAM J. Sci. Statist. Comput., 4, 553-572.
41. Nazareth, L., Computer solution of linear programs: non-simplex algorithms II. Technical Report, Dept. of Pure and Applied Mathematics, Washington State University, Pullman, 1996.
42. Nesterov, Y. E. and Nemirovski, A. S., Interior Point Polynomial Algorithms in Convex Programming. SIAM Publications. SIAM, Philadelphia, USA, 1994.
43. Nesterov, Y. E. and Todd, M. J. (1997). Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22(1), 1-42.
44. Pardalos, P. and Wolkowicz, H. Eds., Topics in Semidefinite and Interior-Point Methods, The Fields Institute for Research in Mathematical Sciences, Communications Series, Providence, RI, 1998. American Mathematical Society.
45. Pataki, G., Cone Programming and Eigenvalue Optimization: Geometry and Algorithms. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1996.
46. Pataki, G. and Tunçel, L., On the generic properties of convex optimization problems in conic form. Technical Report CORR 97-16, Department of Combinatorics and Optimization, Waterloo, Ont, 1997.
47. Powell, M. J. D., The use of band matrices for second derivative approximations in trust region algorithms. In: Advances in Nonlinear Programming (Beijing, 1996), pp. 3-28. Kluwer Acad. Publ., Dordrecht, 1998.
48. Ramana, M. V., An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems. Ph.D. Thesis, Johns Hopkins University, Baltimore, Md, 1993.
49. Ramana, M. V., Tunçel, L. and Wolkowicz, H. (1997). Strong duality for semidefinite programming. SIAM J. Optim., 7(3), 641-662.
50. Rendl, F. (1999). Semidefinite programming and combinatorial optimization. Appl. Numer. Math., 29, 255-281.
51. Rendl, F., Vanderbei, R. J. and Wolkowicz, H. (1995). Max-min eigenvalue problems, primal-dual interior point algorithms, and trust region subproblems. Optim. Methods Softw., 5, 1-16.
52. Rendl, F. and Wolkowicz, H. (1997). A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Programming, 77(2, Ser. B), 273-299.
53. Shapiro, A. (1997). First and second order analysis of nonlinear semidefinite programs. Math. Programming, 77, 301-320.
54. Shapiro, A. and Fan, M. K. H. (1995). On eigenvalue optimization. SIAM J. Optim., 5, 552-569.
55. Sheng, R. and Potra, F. A., Nonsymmetric search directions for semidefinite programming. Technical Report, University of Iowa, Iowa City, IA, 1997.
56. Sorensen, D. C. (1982). Newton's method with a model trust region modification. SIAM J. Numer. Anal., 19, 409-426.
57. Sturm, J. F., Primal-Dual Interior Point Approach to Semidefinite Programming. Ph.D. Thesis, Erasmus University Rotterdam, 1997.
58. Sun, Q, Dealing with dense rows in the solution of sparse linear least squares problems. Technical Report TR95-227, Department of Computer Science, Cornell University, Ithaca, NY, 1995.
59. Terlaky, T. Ed., Interior point methods of mathematical programming. Kluwer Academic Publishers, Dordrecht, 1996.
60. Todd, M. J. (1999). A study of search directions in primal-dual interior-point methods for semidefinite programming. Optim. Methods Softw., 11 and 12, 1-46.
61. Todd, M. J., Toh, K. C. and Tutuncu, R. H., A MATLAB software package for semidefinite programming. Technical Report, School of OR and IE, Cornell University, Ithaca, NY, 1996.
62. Todd, M. J., Toh, K. C. and Tutuncu, R. H. (1998). On the Nesterov-Todd direction in semidefinite programming. SIAM J. Optim., 8(3), 769-796.
63. Toh, K. C., Search directions for primal-dual interior point methods in semidefinite programming. Technical Report, Department of Mathematics, National University of Singapore, Singapore, 1997.
64. Vandenberghe, L. and Boyd, S. (1996). Semidefinite programming. SIAM Rev., 38(1), 49-95.
65. Wolkowicz, H., Saigal, R. and Vandenberghe, L. Eds., Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Boston, MA, 2000. xxvi + 654 pages.
66. Wright, S., Primal-Dual Interior-Point Methods. SIAM, Philadelphia, Pa, 1996.
67. Ye, Y., Interior Point Algorithms: Theory and Analysis. Wiley-Interscience series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.
68. Zhang, Y., Basic equalities and inequalities in primal-dual interior-point methods for semidefinite programming. Technical Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, 1995.
69. Zhao, Q., Semidefinite Programming for Assignment and Partitioning Problems. Ph.D. Thesis, University of Waterloo, 1996.