Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP

Optimization Methods and Software

Volumn 9, Issue 4, 1998, Pages 245-268

  Tseng, Paul ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Convergence; Monotone semi definite LCP; Path following; Search directions

Indexed keywords

COMPUTATIONAL COMPLEXITY; CONVERGENCE OF NUMERICAL METHODS; ITERATIVE METHODS; MATRIX ALGEBRA; PROBLEM SOLVING;

INFEASIBLE PATH FOLLOWING METHODS; MONOTONE SEMIDEFINITE LINEAR COMPLIMENTARY PROBLEMS (LCP); PRIMAL/PRIMAL DUAL/DUAL SEARCH DIRECTIONS; SYMMETRIC BLOCK DIAGONAL MATRIX;

COMPUTATIONAL METHODS;

EID: 0031628254     PISSN: 10556788     EISSN: None     Source Type: Journal    
DOI: 10.1080/10556789808805695     Document Type: Article

References (41)

1. Alizadeh, F. (1992). Optimization over the positive-definite cone: interior point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, edited by P. Pardalos, North-Holland, Amsterdam, 1-25.
2. Alizadeh, F. (1995). Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM J. Optim., 5, 13-51.
3. Alizadeh, F., Haeberly, J.-P. A. and Overton, M. L. (1994). Primal-dual interior point methods for semidefinite programming. Manuscript, Presented at the Math Programming Symposium, Ann Arbor.
4. Alizadeh, F., Haeberly, J.-P. A. and Overton, M. L. (1996). Primal-dual interior point methods for semidefinite programming: convergence rates, stability and numerical results, Working paper, RUTCOR, Rutgers University, New Brunswick.
5. Anstreicher, K. M. and Frampa, M. (1996). A long-step path following algorithm for semidefinite programming problems, Working paper, Department of Management Sciences, University of Iowa, Iowa City.
6. Faybusovich, L. (1995). Semi-definite programming: a path-following algorithm for a linearquadratic functional, Report, Department of Mathematics, University of Notre Dame, Notre Dame, SIAM J. Optimization, to appear.
7. Freund, R. M. (1994). Complexity of an algorithm for finding an approximate solution of a semidefinite program with no regularity assumption, Working paper OR 302-94, Operations Research Center, Massachusetts Institute of Technology, Cambridge.
8. Gonzaga, C. C. (1997). The largest step path following algorithm for monotone linear complementarity problems. Math. Prog., 76, 309-332.
9. Helmberg, C., Rendl, F., Vanderbei, R. J. and Wolkowicz, H. (1996). An interior-point method for semidefinite programming, SIAM J. Optim., 6, 342-361.
10. Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis, Cambridge Press, Cambridge.
11. Jarre, F. (1993). An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31, 1360-1377.
12. Kojima, M., Megiddo, N. and Mizuno, S. (1993). A primal-dual infeasible-interior-point algorithm for linear programming, Math. Programming, 61, 263-280.
13. Kojima, M., Kojima, S. and Hara, S. (1997). Linear algebra for semidefinite programming, RIMS Kokyuroku, Research Institute of Mathematical Sciences, Kyoto University, Kyoto, 1004, 1-23.
14. Kojima, M., Shindoh, S. and Hara, S. (1997). Interior-point methods for the monotone semidefinite linear complementarity problems, SIAM J. Optim., 7, 86-125.
15. Kojima, M., Shida, M. and Shindoh, S. (1995). Local convergence of predictor-corrector infeasible interior-point algorithms for SDPs and SDLCPs, Research Report 306, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Math. Programming, to appear.
16. Kojima, M., Shida, M. and Shindoh, S. (1996). A predictor-corrector interior-point algorithms for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, Research Report 311, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.
17. Kojima, M., Shida, M. and Shindoh, S. (1996). A note on the Nesterov-Todd and the Kojima-Shindo-Hara search directions in semidefinite programming, Research Report 313, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.
18. Lin, C.-J. and Saigal, R. (1995). A predictor-corrector method for semi-definite linear programming, Working paper, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor.
19. Lin, C.-J. and Saigal, R. (1995). An infeasible start predictor-corrector method for semi-definite linear programming, Working paper, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor.
20. Luo, Z.-Q., Sturm, J. F. and Zhang, S. (1996). Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming, Report 9607/A, Econometric Institute, Erasmus University, Rotterdam, SIAM J. Optim., to appear.
21. McShane, K. A. (1994). Superlinearly convergent O(√nL)-iteration interior point algorithms for linear programming and the monotone linear complementarity problem, SIAM J. Optim., 4, 247-261.
22. Mizuno, S., Yoshise, A. and Kikuchi, T. (1989). Practical polynomial time algorithms for linear complementarity problems, J. Oper. Res. Soc. Japan, 32, 75-92.
23. Mizuno, S., Todd, M. J. and Ye, Y. (1993). On adaptive step primal-dual interior-point algorithms for linear programming, Math. Oper. Res., 18, 964-981.
24. Mizuno, S. (1994). Polynomiality of infeasible interior point algorithms for linear programming, Math. Programming, 67, 109-119.
25. Monteiro, R. D. C. (1995). Primal-dual path following algorithms for semidefinite programming, Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, SIAM J. Optim., to appear.
26. Monteiro, R. D. C. and Zhang, Y. (1996). A unified analysis for a class of path-following primaldual interior-point algorithms for semidefinite programming, Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Math. Programming, to appear.
27. Nesterov, Y. E. and Nemirovskii, A. S. (1993). Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia.
28. Nesterov, Y. E. and Todd, M. J. (1997). Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res., 22, 1-42.
29. Nesterov, Y. E. and Todd, M. J. (1995). Primal-dual interior-point methods for self-scaled cones, Technical report 1125, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, revised 1996.
30. Potra, F. and Sheng, R. (1995). A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming, Reports on Computational Mathematics 78, Department of Mathematics, University of Iowa, Iowa City.
31. Potra, F. and Sheng, R. (1995). Homogeneous interior-point algorithms for semidefinite programming. Reports on Computational Mathematics 82, Department of Mathematics, University of Iowa, Iowa City.
32. Potra, F. and Sheng, R. (1996). Superlinear convergence of interior-point algorithms for semidefinite programming, Reports on Computational Mathematics 86, Department of Mathematics, University of Iowa, Iowa City.
33. Shida, M., Shindoh, S. and Kojima, M. (1996). Existence of search directions in interior-point-algorithms for the SDP and the monotone SDLCP, Research Report 310, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.
34. Shida, M. and Shindoh, S. (1996). Monotone semidefinite complementarity problems, Research Report 312, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.
35. Sturm, J. F. and Zhang, S. (1995). Symmetric primal-dual path following algorithms for semidefinite programming, Report 9554/A, Econometric Institute, Erasmus University, Rotterdam, revised 1996.
36. Todd, M. J. Private communication, May (1996).
37. Todd, M. J., Toh, K. C. and Tütüncü, R. H. (1996). On the Nesterov-Todd direction in semidefinite programming, Technical report, School of Operations Research and Industrial Engineering, Cornell University, Ithaca.
38. Tseng, P. (1992). Complexity analysis of a linear complementarity algorithm based on a Lyapunov function, Math. Programming, 53, 297-306.
39. Tseng, P. (1995). Simplified analysis of an O(nL)-iteration infeasible predictor-corrector path following method for monotone LCP, in Recent Trends in Optimization Theory and Applications, edited by R. P. Agarwal, World Scientific, Singapore, 423-434.
40. Vandenberghe, L. and Boyd, S. (1995). A primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, 69, 205-236.
41. Zhang, Y. (1995). On extending primal-dual interior-point algorithms from linear programming to semidefinite programming, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, SIAM J. Optim., to appear.