Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A REGULARIZATION INTERIOR POINT METHOD FOR SEMIDEFINITE PROGRAMMING WITH FREE VARIABLES

  • Liu, Wanxiang (School of Managemen Science, Qufu Normal University) ;
  • Gao, Chengcai (School of Managemen Science, Qufu Normal University) ;
  • Wang, Yiju (School of Managemen Science, Qufu Normal University)
  • Received : 2010.11.20
  • Accepted : 2011.02.07
  • Published : 2011.09.30
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Abstract

In this paper, we proposed a regularization interior point method for semidefinite programming with free variables which can be taken as an extension of the algorithm for standard semidefinite programming. Since an inexact search direction at each iteration is used, the computation of the designed algorithm is much less compared with the existing solution methods. The convergence analysis of the method is established under weak conditions.

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References

  1. F. Alizadeh, Interior-point methods in Semidefinite Programming with applications to com- biatorial optimization, SIAM J. Optim., 5 (1995),13-51. https://doi.org/10.1137/0805002
  2. S. Bellavia, S. Pieraccini, Convergence analysis of an inexact infeasible interior point method for semidefinite programming, Comput. Optim. Appl, 29 (2004),289-313.
  3. C. Helmberg, Semidefinite Programming fo Constrained Optimization, Konrad-Zuse- Zentrum, fur Informationstechnik Berlin, 2000.
  4. E. De. Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, Kluwer Academic Publishers, Dordrecht. The Netherland,2002.
  5. K. Kobayashi, K. Nakata, M. Kojima, A conversition of an SDP having free variables into the standard form SDP, Comput. Optim. Appl. 36 (2007), 289-307. https://doi.org/10.1007/s10589-006-9002-z
  6. M. Kojima, S. Shindoh, S. Hara, Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrics, SIAM J. Optim., 7 (1997), 86-125. https://doi.org/10.1137/S1052623494269035
  7. M. Kojima, M. Shida, S. Shindoh, Local convergence of predictor-corrector infeasible interior-point algorithms for SDPs and SDLCPs, Math. Prog. 80, (1998), 129-160.
  8. J. Korzak, ,Convergence analysis of inexact infeasible-interior-point algorithms for solving linear programming problems, SIAM J. Optim. 11 (2000), 133-148. https://doi.org/10.1137/S1052623497329993
  9. F. A. Miguel, B. Samuel, On handling free variables in interior-point methods for conic linear optimization, SIAM J. Optim., 18 (2007), 1310-1325.
  10. P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Prog. 96 (2003), 293-320. https://doi.org/10.1007/s10107-003-0387-5
  11. G. Pataki, S. Schmieta, The DIMACS Library of Semidefinite-Quadratic-Linear programs, Computational Optimizational Research Center, Columbia University, New York, NY, USA, 1999.
  12. F. A. Potra, R. Sheng, A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming, SIAM J. Optim., 8 (1998), 1007-1028. https://doi.org/10.1137/S1052623495294955
  13. F. A. Potra, R. Sheng, Superlinear convergence of interior-point algorithms for semidefinite programming, J. Optim. Theory Appl. 99 (1998), 103-119. https://doi.org/10.1023/A:1021700210959
  14. F. A. Potra, S. Wright, Interior-point methods, J. Comput. Appl. Math., 124 (2000), 281-302. https://doi.org/10.1016/S0377-0427(00)00433-7
  15. M. Todd, Semidefinite optimization, Acta Numerica, 10 (2001),515-560.
  16. L. Vandenberghe, S. Boyd, Semidefinte programming, SIAM Review, 38 (1996),49-95. https://doi.org/10.1137/1038003
  17. Y. Zhang, On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM J. Optim. 8 (1998),365-386. https://doi.org/10.1137/S1052623495296115
  18. Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representabil- ity, J. Chem. Phys. 120(2004), 261-282.
  19. C. Baiocchi and A. Capelo, Variational and Quasi Variational Inequalities, J. Wiley and Sons, New York, 1984.
  20. D. Chan and J.S. Pang, The generalized quasi variational inequality problems, Math. Oper. Research 7 (1982), 211-222. https://doi.org/10.1287/moor.7.2.211
  21. C. Belly, Variational and Quasi Variational Inequalities, J. Appl.Math. and Computing 6(1999), 234-266.
  22. D. Pang, The generalized quasi variational inequality problems, J. Appl.Math. and Com- puting 8(2002), 123-245.