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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

Yi Xu 1, and  Wenyu Sun 1,

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.
Keywords: global convergence, Nonlinear programming, optimization, successive linear programming, nonlinear semidefinite programming, filter method.
Mathematics Subject Classification: 65k05, 90c3.
Citation: Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and 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), 195-219. doi: 10.1007/s00186-005-0054-0.

[2]

M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," John Wiley & Sons, New York, 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), 161-177.

[4]

R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming, Math. Program., 15 (2004), 303-318.

[5]

B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming, SIAM Journal on Control and Optimization, 40 (2002), 1791-1820. 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), 635-659. doi: 10.1137/S1052623499357258.

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. 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, Technical Report 98/13, Department of Mathematics, University of Namur, Namur, Belgium.

[9]

R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of a Filter-SQP Algorithm, SIAM J. Optim., 13 (2002), 44-59. 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), 341-357. doi: 10.1137/040603851.

[11]

C. Helmberg, Semidefinite programming for combinatorial optimization, Technical Report ZIB-Report ZR-00-34, Konrad-Zuse-Zentrum Berlin, 2000.

[12]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs, Technical Report, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2003.

[13]

F. Jarre, An interior method for nonconvex semidefinite programs, Optimization and Engineering, 1 (2000), 347-372. 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), 251-273. 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), 2341-2361. doi: 10.1007/s11425-009-0168-6.

[16]

W. Miao and W. Sun, A filter-trust-region method for unconstrained optimization, Numerical Mathematics, A Journal of Chinese Universities, 29 (2007), 88-96.

[17]

W. Sun, On filter methods for optimization, The 3rd Australia-China Optimization Workshop, Shanghai University. Invited talk, 2007.

[18]

W. Sun, On filter-type methods for optimization: motivation and development, An invited talk, 4th Sino-Japanese Optimization Meeting (SJOM2008). Cheng Kung University, Taiwan. Aug 26-31, 2008.

[19]

W. Sun and Y. Yuan, "Optimzation Theory and Methods: Nonlinear Programming," Springer, New York, 2006.

[20]

M. J. Todd, Semidefinite optimization, Numerical Mathematics, A Journal of Chinese Universities, 10 (2001), 515-560.

[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. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

[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. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

[23]

R. H. Tutuncu, K. C. Toh and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Math. Prog., 95 (2003), 189-217.

[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), 788-796.

[26]

Y. Zhang, W. Sun and L. Qi, A nonmonotone filter Barzilai-Borwein method for optimization, Asia-Pacific Journal of Operational Research, 27 (2010), 55-69.

show all references

References:
[1]

A. Auslender and H. Ramírez, Penalty and barrier methods for convex semidefinite progranmming, Mathematical Methods of Operations Research, 63 (2006), 195-219. doi: 10.1007/s00186-005-0054-0.

[2]

M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," John Wiley & Sons, New York, 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), 161-177.

[4]

R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming, Math. Program., 15 (2004), 303-318.

[5]

B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming, SIAM Journal on Control and Optimization, 40 (2002), 1791-1820. 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), 635-659. doi: 10.1137/S1052623499357258.

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. 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, Technical Report 98/13, Department of Mathematics, University of Namur, Namur, Belgium.

[9]

R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of a Filter-SQP Algorithm, SIAM J. Optim., 13 (2002), 44-59. 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), 341-357. doi: 10.1137/040603851.

[11]

C. Helmberg, Semidefinite programming for combinatorial optimization, Technical Report ZIB-Report ZR-00-34, Konrad-Zuse-Zentrum Berlin, 2000.

[12]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs, Technical Report, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2003.

[13]

F. Jarre, An interior method for nonconvex semidefinite programs, Optimization and Engineering, 1 (2000), 347-372. 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), 251-273. 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), 2341-2361. doi: 10.1007/s11425-009-0168-6.

[16]

W. Miao and W. Sun, A filter-trust-region method for unconstrained optimization, Numerical Mathematics, A Journal of Chinese Universities, 29 (2007), 88-96.

[17]

W. Sun, On filter methods for optimization, The 3rd Australia-China Optimization Workshop, Shanghai University. Invited talk, 2007.

[18]

W. Sun, On filter-type methods for optimization: motivation and development, An invited talk, 4th Sino-Japanese Optimization Meeting (SJOM2008). Cheng Kung University, Taiwan. Aug 26-31, 2008.

[19]

W. Sun and Y. Yuan, "Optimzation Theory and Methods: Nonlinear Programming," Springer, New York, 2006.

[20]

M. J. Todd, Semidefinite optimization, Numerical Mathematics, A Journal of Chinese Universities, 10 (2001), 515-560.

[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. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

[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. http://www.math.nus.edu.sg/~mattohkc/sdpt3.html.

[23]

R. H. Tutuncu, K. C. Toh and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Math. Prog., 95 (2003), 189-217.

[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), 788-796.

[26]

Y. Zhang, W. Sun and L. Qi, A nonmonotone filter Barzilai-Borwein method for optimization, Asia-Pacific Journal of Operational Research, 27 (2010), 55-69.

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