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2012, 2(1): 193-206. doi: 10.3934/naco.2012.2.193

## A filter successive linear programming method for nonlinear semidefinite programming problems

 1 School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210046, China, China

Received  September 2011 Revised  November 2011 Published  March 2012

In this paper we present a successive linear programming method with filter technique for nonlinear semidefinite programming. Such a method is characterized by use of the dominance concept of multiobjective optimization,~instead of a penalty parameter. The Successive Linear Programming with Filter (SLP-Filter) was used to solve the nonlinear programming (see [8]). In this paper, we extend it to deal with nonlinear semidefinite programming, and prove the convergence of the SLP-Filter for nonlinear semidefinite programming. We report numerical experiments to show the validity of the SLP-Filter method for nonlinear semidefinite programming.
Citation: Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193
##### References:
 [1] A. Auslender and H. Ramírez, Penalty and barrier methods for convex semidefinite progranmming,, Mathematical Methods of Operations Research, 63 (2006), 195. doi: 10.1007/s00186-005-0054-0. [2] M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms,", John Wiley & Sons, (1979). [3] C. Chin and R. Flercher, On the global convergence of an SLP-Filter algorithm that takes EQP steps,, SIAM Journal on Optimization, 96 (2003), 161. [4] R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming,, Math. Program., 15 (2004), 303. [5] B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM Journal on Control and Optimization, 40 (2002), 1791. doi: 10.1137/S0363012900373483. [6] R. Fletcher, N. I. M. Gould, S. Leyffer and A. Wächter, Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming,, SIAM J. Optim., 13 (2002), 635. doi: 10.1137/S1052623499357258. [7] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Mathematical Programming, 91 (2002), 239. doi: 10.1007/s101070100244. [8] R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of an SLP-Filter Algorithm,, Numerical Analysis Report, (). [9] R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of a Filter-SQP Algorithm,, SIAM J. Optim., 13 (2002), 44. doi: 10.1137/S105262340038081X. [10] N. I. M. Gould, C. Sainvitu and Ph. L. Toint, A filter-trust-region method for unconstraint optimization,, SIAM J. Optim., 16 (2005), 341. doi: 10.1137/040603851. [11] C. Helmberg, Semidefinite programming for combinatorial optimization,, Technical Report ZIB-Report ZR-00-34, (2000), 00. [12] X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs,, Technical Report, (2003). [13] F. Jarre, An interior method for nonconvex semidefinite programs,, Optimization and Engineering, 1 (2000), 347. doi: 10.1023/A:1011562523132. [14] C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Succseeive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251. doi: 10.1007/s10589-005-3231-4. [15] C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, Science in China Series A, 52 (2009), 2341. doi: 10.1007/s11425-009-0168-6. [16] W. Miao and W. Sun, A filter-trust-region method for unconstrained optimization,, Numerical Mathematics, 29 (2007), 88. [17] W. Sun, On filter methods for optimization,, The 3rd Australia-China Optimization Workshop, (2007). [18] W. Sun, On filter-type methods for optimization: motivation and development,, An invited talk, (2008), 26. [19] W. Sun and Y. Yuan, "Optimzation Theory and Methods: Nonlinear Programming,", Springer, (2006). [20] M. J. Todd, Semidefinite optimization,, Numerical Mathematics, 10 (2001), 515. [21] K. C. Toh, R. H. Tutuncu and M. J. Todd, SDPT3 version 4.0 (beta)- a MATLAB software for semidefinite-quadratic-linear programming,, updated in 17 July, (2006). [22] K. C. Toh, R. H. Tutuncu and M. J. Todd, On the implementation and usage of SDPT3 - a MATLAB software package for semidefinite-quadratic-linear programming version 4.0,, 17 July, (2006). [23] R. H. Tutuncu, K. C. Toh and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3,, Math. Prog., 95 (2003), 189. [24] H. Wolkowicz, R. Saigal and L. Vandenberghe, "Handbook of Semidefinite Programming,", Boston: Kluwer Academic Publishers, (2000). [25] Z. Yang, W. Sun and L. Qi, On global convergence of a filter-trust-region algorithm for solving nonsmooth equations,, International Journal of Computer Mathematics, 87 (2010), 788. [26] Y. Zhang, W. Sun and L. Qi, A nonmonotone filter Barzilai-Borwein method for optimization,, Asia-Pacific Journal of Operational Research, 27 (2010), 55.

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##### References:
 [1] A. Auslender and H. Ramírez, Penalty and barrier methods for convex semidefinite progranmming,, Mathematical Methods of Operations Research, 63 (2006), 195. doi: 10.1007/s00186-005-0054-0. [2] M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms,", John Wiley & Sons, (1979). [3] C. Chin and R. Flercher, On the global convergence of an SLP-Filter algorithm that takes EQP steps,, SIAM Journal on Optimization, 96 (2003), 161. [4] R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming,, Math. Program., 15 (2004), 303. [5] B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM Journal on Control and Optimization, 40 (2002), 1791. doi: 10.1137/S0363012900373483. [6] R. Fletcher, N. I. M. Gould, S. Leyffer and A. Wächter, Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming,, SIAM J. Optim., 13 (2002), 635. doi: 10.1137/S1052623499357258. [7] R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Mathematical Programming, 91 (2002), 239. doi: 10.1007/s101070100244. [8] R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of an SLP-Filter Algorithm,, Numerical Analysis Report, (). [9] R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of a Filter-SQP Algorithm,, SIAM J. Optim., 13 (2002), 44. doi: 10.1137/S105262340038081X. [10] N. I. M. Gould, C. Sainvitu and Ph. L. Toint, A filter-trust-region method for unconstraint optimization,, SIAM J. Optim., 16 (2005), 341. doi: 10.1137/040603851. [11] C. Helmberg, Semidefinite programming for combinatorial optimization,, Technical Report ZIB-Report ZR-00-34, (2000), 00. [12] X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs,, Technical Report, (2003). [13] F. Jarre, An interior method for nonconvex semidefinite programs,, Optimization and Engineering, 1 (2000), 347. doi: 10.1023/A:1011562523132. [14] C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Succseeive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251. doi: 10.1007/s10589-005-3231-4. [15] C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, Science in China Series A, 52 (2009), 2341. doi: 10.1007/s11425-009-0168-6. [16] W. Miao and W. Sun, A filter-trust-region method for unconstrained optimization,, Numerical Mathematics, 29 (2007), 88. [17] W. Sun, On filter methods for optimization,, The 3rd Australia-China Optimization Workshop, (2007). [18] W. Sun, On filter-type methods for optimization: motivation and development,, An invited talk, (2008), 26. [19] W. Sun and Y. Yuan, "Optimzation Theory and Methods: Nonlinear Programming,", Springer, (2006). [20] M. J. Todd, Semidefinite optimization,, Numerical Mathematics, 10 (2001), 515. [21] K. C. Toh, R. H. Tutuncu and M. J. Todd, SDPT3 version 4.0 (beta)- a MATLAB software for semidefinite-quadratic-linear programming,, updated in 17 July, (2006). [22] K. C. Toh, R. H. Tutuncu and M. J. Todd, On the implementation and usage of SDPT3 - a MATLAB software package for semidefinite-quadratic-linear programming version 4.0,, 17 July, (2006). [23] R. H. Tutuncu, K. C. Toh and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3,, Math. Prog., 95 (2003), 189. [24] H. Wolkowicz, R. Saigal and L. Vandenberghe, "Handbook of Semidefinite Programming,", Boston: Kluwer Academic Publishers, (2000). [25] Z. Yang, W. Sun and L. Qi, On global convergence of a filter-trust-region algorithm for solving nonsmooth equations,, International Journal of Computer Mathematics, 87 (2010), 788. [26] Y. Zhang, W. Sun and L. Qi, A nonmonotone filter Barzilai-Borwein method for optimization,, Asia-Pacific Journal of Operational Research, 27 (2010), 55.
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