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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 & 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.  Google Scholar

[2]

M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," John Wiley & Sons, New York, 1979.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. doi: 10.1007/s101070100244.  Google Scholar

[8]

R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of an SLP-Filter Algorithm,, Numerical Analysis Report, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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. Google Scholar

[13]

F. Jarre, An interior method for nonconvex semidefinite programs, Optimization and Engineering, 1 (2000), 347-372. doi: 10.1023/A:1011562523132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

H. Wolkowicz, R. Saigal and L. Vandenberghe, "Handbook of Semidefinite Programming," Boston: Kluwer Academic Publishers, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. S. Bazaraa and C. M. Shetty, "Nonlinear Programming Theory and Algorithms," John Wiley & Sons, New York, 1979.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. doi: 10.1007/s101070100244.  Google Scholar

[8]

R. Fletcher, S. Leyffer and Ph.L. Toint, On the global convergence of an SLP-Filter Algorithm,, Numerical Analysis Report, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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. Google Scholar

[13]

F. Jarre, An interior method for nonconvex semidefinite programs, Optimization and Engineering, 1 (2000), 347-372. doi: 10.1023/A:1011562523132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

H. Wolkowicz, R. Saigal and L. Vandenberghe, "Handbook of Semidefinite Programming," Boston: Kluwer Academic Publishers, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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