Interior point methods for nonlinear optimization

Lecture Notes in Mathematics

Volumn 1989, Issue , 2010, Pages 215-276

  Pólik, Imre ^a   Terlaky, Tamás ^a  

^a Lehigh University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77950107851     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-11339-0_4     Document Type: Article

References (100)

1. F. Alizadeh. Combinatorial optimization with interior point methods and semidefinite matrices, Ph.D. thesis, University of Minnesota, Minneapolis, USA, 1991.
2. F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM Journal on Optimization, 5(1),13-51, 1995.
3. F. Alizadeh and D. Goldfarb. Second-order cone programming, Mathematical Programming, Series B, 95,3-51, 2002.
4. F. Alizadeh and S. H. Schmieta. Symmetric cones, potential reduction methods, In H.Wolkowicz, R. Saigal, and L. Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 8, pages 195-233. Kluwer Academic Publishers, 2000.
5. E. D. Andersen, J. Gondzio, Cs. Ḿesźaros, and X. Xu. Implementation of interior point methods for large scale linear programming, In T. Terlaky, editor, Interior Point Methods of Mathematical Programming, pages 189-252. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
6. E. D. Andersen, B. Jensen, R. Sandvik, and U.Worsoe. The improvements in MOSEK version 5., Technical report 1-2007, MOSEK ApS, Fruebjergvej 3 Box 16, 2100 Copenhagen, Denmark, 2007.
7. E. D. Andersen and Y. Ye. A computational study of the homogeneous algorithm for large-scale convex optimization, Computational Optimization and Applications, 10(3), 243-280, 1998.
8. E. D. Andersen and Y. Ye. On a homogeneous algorithm for the monotone complementarity problem, Mathematical Programming, Series A, 84(2),375-399, 1999.
9. K. Anstreicher. Large step volumetric potential reduction algorithms for linear programming, Annals of Operations Research, 62, 521-538, 1996.
10. K. Anstreicher. Volumetric path following algorithms for linear programming, Mathematical Programming, 76, 245-263, 1997.
11. A. Ben-Tal and A. Nemirovski Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series on Optimization. SIAM, Philadelphia, PA, 2001.
12. R. E. Bixby. Solving real-world linear programs: A decade and more of progress, Operations Research, 50(1), 3-15, 2002.
13. B. Borchers and J. G. Young. Implementation of a primal-dual method for SDP on a shared memory parallel architecture, Computational Optimization and Applications, 37(3), 355-369, 2007.
14. R. Byrd, J. Nocedal, and R.Waltz. KNITRO: An integrated package for nonlinear optimization, In G. Di Pillo and M. Roma, editors, Large-Scale Nonlinear Optimization, volume 83 of Nonconvex Optimization and Its Applications. Springer, 2006.
15. M. Colombo and J. Gondzio. Further development of multiple centrality correctors for interior point methods, Computational Optimization and Applications, 41(3), 277-305, 2008.
16. E. de Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, Kluwer Academic Publichers, Dordrecht, The Netherlands, 2002.
17. E. de Klerk, C. Roos, and T. Terlaky. Semi-definite problems in truss topology optimization, Technical Report 95-128, Faculty of Technical Mathematics and Informatics, T.U. Delft, The Netherlands, 1995.
18. E. de Klerk, C. Roos, and T. Terlaky. Initialization in semidefinite programming via a self-dual, skew-symmetric embedding, OR Letters, 20, 213-221, 1997.
19. E. de Klerk, C. Roos, and T. Terlaky. Infeasible-start semidefinite programming algorithms via self-dual embeddings. In P. M. Pardalos and H. Wolkowicz, editors, Topics in Semidefinite and Interior Point Methods" volume 18 of Fields Institute Communications, pages 215-236. AMS, Providence, RI, 1998.
20. E. de Klerk, C. Roos, and T. Terlaky. On primal-dual path-following algorithms for semidefinite programming, In F. Gianessi, S. Komĺosi, and T. Rapcśak, editors, New Trends in Mathematical Programming, pages 137-157. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
21. A. Deza, E. Nematollahi, and T. Terlaky. How good are interior point methods? Klee- Minty cubes tighten iteration-complexity bounds, Mathematical Programming, 113, 1-14, 2008.
22. J. Faraut and A. Koŕanyi Analysis on Symmetric Cones, Oxford Mathematical Monographs. Oxford University Press, 1994.
23. A. V. Fiacco and G. P. McCormick Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, 1968. Reprint: Vol.4. SIAM Classics in Applied Mathematics, SIAM Publications, Philadelphia, USA, 1990.
24. J. Forrest, D. de le Nuez, and R. Lougee-Heimer CLP User Guide, IBM Corporation, 2004.
25. A. Forsgren, Ph. E. Gill, and M. H. Wright. Interior methods for nonlinear optimization, SIAM Review, 44(4), 525-597, 2002.
26. R. M. Freund. On the behavior of the homogeneous self-dual model for conic convex optimization, Mathematical Programming, 106(3), 527-545, 2006.
27. R. Frisch. The logarithmic potential method for solving linear programming problems, Memorandum, University Institute of Economics, Oslo, Norway, 1955.
28. GNU Linear Programming Kit Reference Manual, Version 4.28, 2008.
29. M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM, 42, 1115-1145, 1995.
30. D. Goldfarb and K. Scheinberg. A product-form cholesky factorization method for handling dense columns in interior point methods for linear programming, Mathematical Programming, 99(1), 1-34, 2004.
31. D. Goldfarb and K. Scheinberg. Product-form cholesky factorization in interior point methods for second-order cone programming, Mathematical Programming, 103(1), 153-179, 2005.
32. A. J. Goldman and A. W. Tucker. Theory of linear programming, In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems, number 38 in Annals of Mathematical Studies, pages 53-97. Princeton University Press, Princeton, New Jersey, 1956.
33. J. Gondzio. Warm start of the primal-dual method applied in the cutting plane scheme, Mathematical Programming, 83(1), 125-143, 1998.
34. J. Gondzio and A. Grothey. Direct solution of linear systems of size 109 arising in optimization with interior point methods, In R. Wyrzykowski, J. Dongarra, N. Meyer, and J. Wasniewski, editors, Parallel Processing and Applied Mathematics, number 3911 in Lecture Notes in Computer Science, pages 513-525. Springer-Verlag, Berlin, 2006.
35. J. Gondzio and A. Grothey. A new unblocking technique to warmstart interior point methods based on sensitivity analysis, SIAM Journal on Optimization, 3, 1184-1210, 2008.
36. J. Gondzio and T. Terlaky. A computational view of interior point methods for linear programming, In J. E. Beasley, editor, Advances in Linear and Integer Programming, pages 103-185. Oxford University Press, Oxford, 1996.
37. I. S. Gradshteyn and I. M. Ryzhik Tables of Integrals, Series, and Products, Academic Press, San Diego, CA, 6th edition, 2000.
38. M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, Web page and software, http://stanford.edu/?boyd/cvx, 2008.
39. M. Grant and S. Boyd. Graph implementations for nonsmooth convex programs, In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control (a tribute to M. Vidyasagar). Springer, 95-110, 2008.
40. M. Grötschel, L. Lov́asz, and A. Schrijver Geometric Algorithms and Combinatorial Optimization. Springer Verlag, 1988.
41. Ch. Gúeret, Ch. Prins, and M. Sevaux Applications of optimization with Xpress-MP, Dash Optimization, 2002. Translated and revised by Susanne Heipcke.
42. D. den Hertog Interior Point Approach to Linear, Quadratic and Convex Programming, volume 277 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
43. T. Ilĺes and T. Terlaky. Pivot versus interior point methods: Pros and cons, European Journal of Operational Research, 140(2), 170-190, 2002.
44. I. D. Ivanov and E. de Klerk. Parallel implementation of a semidefinite programming solver based on CSDP in a distributed memory cluster, Optimization Methods and Software, 25(3), 405-420, 2010.
45. B. Jansen, Interior Point Techniques in Optimization. Complexity, Sensitivity and Algorithms, volume 6 of Applied Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
46. B. Jansen, J. J. de Jong, C. Roos, and T. Terlaky. Sensitivity analysis in linear programming: just be careful!, European Journal of Operations Research, 101(1), 15-28, 1997.
47. B. Jansen, C. Roos, and T. Terlaky. The theory of linear programming: Skew symmetric self-dual problems and the central path, Optimization, 29, 225-233, 1994.
48. N. K. Karmarkar. A new polynomial-time algorithm for linear programming, Combinatorica, 4, 373-395, 1984.
49. M. Kočvara and M. Stingl PENSDP Users Guide (Version 2.2), PENOPT GbR, 2006.
50. J. Löfberg. YALMIP: a toolbox for modeling and optimization in MATLAB, In Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pages 284-289, 2004.
51. L. Lov́asz and A. Schrijver. Cones of matrices and setfunctions, and 0-1 optimization, SIAM Journal on Optimization, 1(2), 166-190, 1991.
52. Z.-Q. Luo, J. F. Sturm, and S. Zhang. Conic linear programming and self-dual embedding, Optimization Methods and Software, 14, 169-218, 2000.
53. K. A. McShane, C. L. Monma, and D. F. Shanno. An implementation of a primaldual interior point method for linear programming, ORSA Journal on Computing, 1, 70-83, 1989.
54. N. Megiddo. Pathways to the optimal set in linear programming, In N. Megiddo, editor, Progress in Mathematical Programming: Interior Point and Related Methods, pages 131-158. Springer Verlag, New York, 1989. Identical version in: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, pages 1-35, 1986.
55. S. Mehrotra. On the implementation of a (primal-dual) interior point method, SIAM Journal on Optimization, 2(4), 575-601, 1992.
56. S. Mehrotra and Y. Ye. On finding the optimal facet of linear programs, Mathematical Programming, 62, 497-515, 1993.
57. R. D. C. Monteiro. Primal-dual path-following algorithms for semidefinite programming, SIAM Journal on Optimization, 7, 663-678, 1997.
58. R. D. C. Monteiro and S. Mehrotra. A general parametric analysis approach and its implications to sensitivity analysis in interior point methods, Mathematical Programming, 72, 65-82, 1996.
59. R. D. C. Monteiro and M. J. Todd. Path-following methods. In H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, chapter 10, pages 267-306. Kluwer Academic Publishers, 2000.
60. R. D. C.Monteiro and T. Tsuchiya. Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions, Mathematical Programming, 88(1), 61-83, 2000.
61. K. Nakata, M. Yamashita, K. Fujisawa, and M. Kojima. A parallel primal-dual interior-point method for semidefinite programs using positive definite matrix completion, Parallel Computing, 32, 24-43, 2006.
62. A. S. Nemirovski and M. J. Todd. Interior-point methods for optimization, Acta Numerica, 17, 191-234, 2008.
63. Y. E. Nesterov and A. Nemirovski Interior-Point Polynomial Algorithms in Convex Programming, volume 13 of SIAM Studies in Applied Mathematics. SIAM Publications, Philadelphia, PA, 1994.
64. Y. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research, 22(1), 1-42, 1997.
65. M. L. Overton. On minimizing the maximum eigenvalue of a symmetric matrix, SIAM Journal on Matrix Analysis and Applications, 9(2), 256-268, 1988.
66. J. Peng, C. Roos, and T. Terlaky. New complexity analysis of the primal-dual Newton method for linear optimization, Annals of Operations Research, 99, 23-39, 2000.
67. J. Peng, C. Roos, and T. Terlaky Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton, NJ, 2002.
68. L. Porkoĺab and L. Khachiyan. On the complexity of semidefinite programs, Journal of Global Optimization, 10(4), 351-365, 1997.
69. F. Potra and R. Sheng. On homogeneous interior-point algorithms for semidefinite programming, Optimization Methods and Software, 9(3), 161-184, 1998.
70. F. Potra and Y. Ye. Interior-point methods for nonlinear complementarity problems, Journal of Optimization Theory and Applications, 68, 617-642, 1996.
71. A. U. Raghunathan and L. T. Biegler. An interior point method for mathematical programs with complementarity constraints (MPCCs), SIAM Journal on Optimization, 15(3), 720-750, 2005.
72. C. Roos, T. Terlaky, and J.-Ph. Vial Theory and Algorithms for Linear Optimization. An Interior Approach, Springer, New York, USA, 2nd edition, 2006.
73. S. H. Schmieta and F. Alizadeh. Associative and Jordan algebras, and polynomial time interior-point algorithms, Mathematics of Operations Research, 26(3), 543-564, 2001.
74. Gy. Sonnevend. An "analytic center" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, In A. Pŕekopa, J. Szelezśan, and B. Strazicky, editors, System Modeling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, volume 84 of Lecture Notes in Control and Information Sciences, pages 866-876. Springer Verlag, Berlin, West Germany, 1986.
75. J. F. Sturm. Primal-dual interior point approach to semidefinite programming, In J. B. G. Frenk, C. Roos, T. Terlaky, and S. Zhang, editors, High Performance Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
76. J. F. Sturm. Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones, Optimization Methods and Software, 11-12, 625-653, 1999.
77. J. F. Sturm. Implementation of interior point methods for mixed semidefinite and second order cone optimization problems, Optimization Methods and Software, 17(6), 1105-1154, 2002.
78. J. F. Sturm. Avoiding numerical cancellation in the interior point method for solving semidefinite programs, Mathematical Programming, 95(2), 219-247, 2003.
79. J. F. Sturm and S. Zhang. An interior point method, based on rank-1 updates, for linear programming, Mathematical Programming, 81, 77-87, 1998.
80. T. Terlaky, editor Interior Point Methods of Mathematical Programming, volume 5 of Applied Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
81. T. Terlaky. An easy way to teach interior-point methods, European Journal of Operational Research, 130(1), 1-19, 2001.
82. M. J. Todd. A study of search directions in primal-dual interior-point methods for semidefinite programming, Optimization Methods and Software, 11, 1-46, 1999.
83. K. C. Toh, R. H. Tütüncü, and M. J. Todd. On the implementation of SDPT3 (version 3.1) - a Matlab software package for semidefinite-quadratic-linear programming, In Proceedings of the IEEE Conference on Computer-Aided Control System Design, 2004.
84. T. Tsuchiya. A convergence analysis of the scaling-invariant primal-dual pathfollowing algorithms for second-order cone programming, Optimization Methods and Software, 11, 141-182, 1999.
85. A. W. Tucker. Dual systems of homogeneous linear relations, In H. W. Kuhn and A. W. Tucker, editors, Linear Inequalities and Related Systems, number 38 in Annals of Mathematical Studies, pages 3-18. Princeton University Press, Princeton, New Jersey, 1956.
86. R. H. Tütüncü, K. C. Toh, and M. J. Todd. Solving semidefinite-quadratic-linear programs using SDPT3, Mathematical Programming Series B, 95, 189-217, 2003.
87. P. M. Vaidya. A new algorithm for minimizing convex functions over convex sets, Mathematical Programming, 73(3), 291-341, 1996.
88. L. Vandenberghe and S. Boyd. Semidefinite programming, SIAM Review, 38, 49-95, 1996.
89. R. J. Vanderbei LOQO User,s Guide - Version 4.05, Princeton University, School of Engineering and Applied Science, Department of Operations Research and Financial Engineering, Princeton, New Jersey, 2006.
90. A. Wächter and L. T. Biegler. On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106(1), 25-57, 2006.
91. H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, 2000.
92. M. H. Wright. The interior-point revolution in optimization: History, recent developments, and lasting consequences, Bulletin (New Series) of the AmericanMathematical Society, 42(1), 39-56, 2004.
93. S. J. Wright. Primal-Dual Interior-Point Methods, SIAM, Philadelphia, USA, 1997.
94. X. Xu, P. Hung, and Y. Ye. A simplified homogeneous and self-dual linear programming algorithm and its implementation, Annals of Operations Research, 62, 151-172, 1996.
95. M. Yamashita, K. Fujisawa, and M. Kojima. Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0), Optimization Methods and Software, 18, 491-505, 2003.
96. Y. Ye Interior-Point Algorithms: Theory and Analysis, Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley and Sons, New York, 1997.
97. Y. Ye, M. J. Todd, and S. Mizuno. An O( √ nL)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research, 19, 53-67, 1994.
98. E. A. Yildirim and S. J. Wright. Warm start strategies in interior-point methods for linear programming, SIAM Journal on Optimization, 12(3), 782-810, 2002.
99. Y. Zhang. On extending primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM Journal of Optimization, 8, 356-386, 1998.
100. Y. Zhang. Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optimization Methods Software, 10, 1-31, 1998.