Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones

SIAM Journal on Optimization

Volumn 17, Issue 4, 2006, Pages 1129-1153

  Yoshise, Akiko ^a  

^a UNIVERSITY OF TSUKUBA   (Japan)

Author keywords

Complementarity problem; Detecting infeasibility; Existence of trajectory; Homogeneous algorithm; Interior point method; Symmetric cone

Indexed keywords

COMPUTATIONAL COMPLEXITY; MATHEMATICAL MODELS; MATRIX ALGEBRA; OPTIMIZATION; PROBLEM SOLVING;

COMPLEMENTARITY PROBLEM; HOMOGENEOUS ALGORITHM; OPTIMIZATION THEORY; SYMMETRIC CONES;

NONLINEAR SYSTEMS;

EID: 36248962181     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/04061427X     Document Type: Article

References (33)

1. F. ALIZADEH AND S. H. SCHMIETA, Symmetric cones, potential reduction methods and wordby-word extensions, in Handbook of Semidefinite Programming. Theory, Algorithms, and Applications, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Kluwer Academic Publishers, Boston, MA, 2000, pp. 195-233.
2. E. ANDERSEN AND Y. YE, On a homogeneous algorithm for the monotone complementarity problems, Math. Program., 84 (1999), pp. 375-400.
3. C. B. CHUA, A New Notion of Weighted Centers for Semidefinite Programming, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada, 2004.
4. E. DE KLERK, C. ROOS, AND T. TERLAKY, Initialization in semidefinite programming via a self-dual skew-symmetric embedding, Oper. Res. Lett., 20 (1997), pp. 213-221.
5. J. FARAUT AND A. KORÁNYI, Analysis on Symmetric Cones, Oxford Science Publishers, Oxford, UK, 1994.
6. L. FAYBUSOVICH, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), pp. 331-357.
7. L. FAYBUSOVICH, Linear systems in Jordan algebras and primal-dual interior point algorithms, J. Comput. Appl. Math., 86 (1997), pp. 149-175.
8. M. S. GOWDA, R. SZNAJDER, AND J. TAO, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra Appl., 393 (2004), pp. 203-232.
9. O. GÜLER, Existence of interior points and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), pp. 128-148.
10. O. GÜLER, Barrier functions in interior-point methods, Math. Oper. Res., 21 (1996), pp. 860-885.
11. R. A. HAUSER AND O. GÜLER, Self-scaled barrier functions on symmetric cones and their classification, Found. Comput. Math., 2 (2002), pp. 121-143.
12. S. HAYASHI, N. YAMASHITA, AND M. FUKUSHIMA, Robust Nash equilibria and second-order cone complementarity problems, J. Nonlinear Convex Anal., 6 (2005), pp. 283-296.
13. M. KOJIMA, N. MEGIDDO, AND T. NOMA, Homotopy continuation methods for nonlinear complementarity problems, Math. Oper. Res., 16 (1991), pp. 754-774.
14. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming, N. Megiddo, ed., Springer-Verlag, New York, 1989, pp. 29-47.
15. M. KOJIMA, S. SHINDOH, AND S. HARA, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices, SIAM J. Optim., 7 (1997), pp. 86-125.
16. Z.-Q. LUO, J. F. STURM, AND S. ZHANG, Conic convex programming and self-dual embedding, Optim. Methods Softw., 14 (2000), pp. 196-218.
17. R.D.C. MONTEIRO AND I. ADLER, Interior path following primal-dual algorithms. I. Linear programming, Math. Programming, 44 (1989), pp. 27-41.
18. R.D.C. MONTEIRO AND J.-S. PANG, Properties of an interior-point mapping for mixed complementarity problems, Math. Oper. Res., 21 (1996), pp. 629-654.
19. R.D.C. MONTEIRO AND J.-S. PANG, On two interior-point mappings for nonlinear semidefinite complementarity problems, Math. Oper. Res., 23 (1998), pp. 39-60.
20. R.D.C. MONTEIRO AND P. ZANJACOMO, General interior-point maps and existence of weighted paths for nonlinear semidefinite complementarity problems, Math. Oper. Res., 25 (2000), pp. 381-399.
21. YU. E. NESTEROV AND M. J. TODD, Self-scaled barriers and interior-point for convex programming,Math. Oper. Res., 22 (1997), pp. 1-42.
22. YU. E. NESTEROV AND M. J. TODD, Primal-dual interior-point methods for self-scaled cones,SIAM J. Optim., 8 (1998), pp. 324-364.
23. YU. E. NESTEROV, M. J. TODD, AND Y. YE, Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems,Math. Program., 84 (1999), pp. 227-267.
24. B. K. RANGARAJAN, Polynomial convergence of infeasible-interior-point methods over symmetric cones, SIAM J. Optim., 16 (2006), pp. 1211-1229.
25. B. K. RANGARAJAN AND M. J. TODD, Convergence of Infeasible-Interior-Point Methods for Self-Scaled Conic Programming, Technical report TR1388, Department of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 2003.
26. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer-Verlag, New York, 1998.
27. S. H. SCHMIETA, Complete Classification of Self-Scaled Barrier Functions, Technical report, Department of IEOR, Columbia University, New York, NY, 2000.
28. S. H. SCHMIETA AND F. ALIZADEH, Extension of primal-dual interior-point algorithm to symmetric cones, Math. Program., 96 (2003), pp. 409-438.
29. M. SHIDA, S. SHINDOH, AND M. KOJIMA, Centers of monotone generalized complementarity problems, Math. Oper. Res., 22 (1997), pp. 969-976.
30. J. F. STURM, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11-12 (1999), pp. 625-653.
31. J. TAO AND M. S. GOWDA, Some P-Properties for Nonlinear Transformations on Euclidean Jordan Algebras, Technical report TRGOW04-01, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD, 2004.
32. Y. YE, On homogeneous and self-dual algorithms for LCP, Math. Programming, 76 (1997), pp. 211-221.
33. 0 and P* Nonlinear Complementarity Problems, DPS 1059, Institute of Policy and Planning Sciences, The University of Tsukuba, Tsukuba, Japan, 2003.