Self-regular functions and new search directions for linear and semidefinite optimization

Mathematical Programming, Series B

Volumn 93, Issue 1, 2002, Pages 129-171

  Peng, Jiming ^a   Roos, Cornelis ^b   Terlaky, Tamás ^a  

^a MCMASTER UNIVERSITY   (Canada)

^b DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Interior point method; Linear optimization; Polynomial complexity; Primal dual method; Self regularity; Semidefinite optimization

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; ITERATIVE METHODS; LINEAR SYSTEMS; OPTIMIZATION; POLYNOMIALS;

INTERIOR-POINT METHOD; LINEAR OPTIMIZATION; POLYNOMIAL COMPLEXITY; PRIMAL-DUAL METHOD; SELF-REGULARITY; SEMIDEFINITE OPTIMIZATION;

FUNCTIONS;

EID: 33750979581     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s101070200296     Document Type: Article

References (26)

1. Andersen, E.D., Gondzio, J., Mészáros, Cs., Xu., X. (1996): Implementation of interior point methods for large scale linear programming. In: Terlaky, T., ed., Interior point methods of mathematical programming, pp. 189-252. Kluwer Academic Publishers, Dordrecht, The Netherlands
2. Bellman, R. (1995): Introduction to matrix analysis. Vol. 12 Classics in Applied Mathematics. SIAM, Philadelphia
3. Horn, R.A., Johnson, C.R. (1991): Topics in Matrix Analysis. Cambridge University Press
4. de Klerk, E. (1997): Interior Point Methods for Semidefinite Programming. Ph.D. Thesis, Faculty of ITS/TWI, Delft University of Technology, The Netherlands
5. Hung, P., Ye, Y. (1996): An asymptotically O(√nL)-iteration path-following linear programming algorithm that uses long steps. SIAM J. Optimization 6, 570-586
6. Jansen, B., Roos, C., Terlaky, T., Ye, Y. (1997): Improved complexity using higher order correctors for primal-dual Dikin affine scaling. Mathematical Programming Series B 76, 117-130
7. Karmarkar, N.K. (1984): A new polynomial-time algorithm for linear programming. Combinatorica 4, 373-395
8. Megiddo, N. (1989): 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, New York
9. Mehrotra, S., (1992): On the implementation of a (primal-dual) interior point method, SIAM J. Optimization, 2(4), 575-601
10. Mehrotra, S., Ye, Y. (1993): On finding the optimal facet of linear programs. Mathematical Programming 62, 497-515
11. Monteiro, R.D.C., Adler, I., Resende, M.G.C. (1990): A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programmins and its power series. Mathematics of Operations Research, 15, 191-214
12. Nesterov, Y.E., Nemirovskii, A.S. (1994): Interior point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics. Vol. 13 SIAM, Philadelphia, USA
13. Nesterov, Y.E., Todd, M.J. (1997): Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research 22(1), 1-42
14. Peng, J., Roos, C., Terlaky, T. (2002): A new class of polynomial primal-dual methods for linear and semidefinite optimization. To appear in European Journal of Operations Research
15. Peng, J., Roos, C., Terlaky, T. (2000): New polynomial primal-dual methods for second-order conic optimization based on self-regular proximities. To appear in SIAM J. Optimization http://www.cas.mcmaster.ca/~oplab/publication/ report/2000-5.ps
16. Peng, J., Roos, C., Terlaky, T., Yoshise, A. (2000): New polynomial methods for nonlinear complementarity problems based on self-regular proximities. Technical Report 2000-6, AdvOL, Dept. CAS, McMaster University, Hamilton, ON, Canada. http://www.cas.mcmaster.ca/~oplab/publication/report/2000- 6.ps
17. Roos, C., Terlaky, T., Vial, J.-Ph. (1997): Theory and Algorithms for Linear Optimization. An Interior Approach. John Wiley & Sons, Chichester, UK
18. Rudin, W. (1978): Principles of Mathematical Analysis. Mac-Graw Hill Company, New York
19. Sonnevend, G. (1986): An "analytic center" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prékopa, A., Szelezsán, J., Strazicky, B. (eds.): System Modelling and Optimization : Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985. Vol. 84 of Lecture Notes in Control and Information Sciences, pp. 866-876. Springer, Berlin, West-Germany
20. Sturm, J.F., Zhang, S. (1999): Symmetric primal-dual path following algorithms for semidefinite programming. Applied Numerical Mathematics 29, 301-315
21. Todd, M.J. (1999): A study of search directions in primal-dual interior-point methods for semidefinite programming. Optimization Methods and Software 11, 1-46
22. Todd, M.J., Toh, K.C., Tütüncü, R.H. (1998): On the Nesterov-Todd direction in semidefinite programming. SIAM J. Optimization 8, 769-796
23. Wright, S.J. (1977): Primal-Dual Interior-Point Methods. SIAM, Philadelphia, USA
24. Ye, Y. (1992): On the finite convergence of interior-point algorithms for linear programming, Mathematical Programming 57, 325-335
25. Ye, Y. (1997): Interior Point Algorithms, Theory and Analysis. John Wiley & Sons, Chichester, UK
26. Zhang, L.L., Zhang, Y. (1995): On polynomiality of the Mehrotra-type predictor-corrector interior-point algorithms. Mathematical Programming 68, (1995) 303-318