• Previous Article
    Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences
  • NACO Home
  • This Issue
  • Next Article
    On some inverse singular value problems with Toeplitz-related structure
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.  Google Scholar

[2]

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

[4]

R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming,, Math. Program., 15 (2004), 303.   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.  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.  doi: 10.1137/S1052623499357258.  Google Scholar

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Mathematical Programming, 91 (2002), 239.  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.  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.  doi: 10.1137/040603851.  Google Scholar

[11]

C. Helmberg, Semidefinite programming for combinatorial optimization,, Technical Report ZIB-Report ZR-00-34, (2000), 00.   Google Scholar

[12]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs,, Technical Report, (2003).   Google Scholar

[13]

F. Jarre, An interior method for nonconvex semidefinite programs,, Optimization and Engineering, 1 (2000), 347.  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.  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.  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, 29 (2007), 88.   Google Scholar

[17]

W. Sun, On filter methods for optimization,, The 3rd Australia-China Optimization Workshop, (2007).   Google Scholar

[18]

W. Sun, On filter-type methods for optimization: motivation and development,, An invited talk, (2008), 26.   Google Scholar

[19]

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

[20]

M. J. Todd, Semidefinite optimization,, Numerical Mathematics, 10 (2001), 515.   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).   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).   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.   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.   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.   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.  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, (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.   Google Scholar

[4]

R. Correa and H. Ramírez, A global algorithm for nonlinear semidefinite programming,, Math. Program., 15 (2004), 303.   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.  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.  doi: 10.1137/S1052623499357258.  Google Scholar

[7]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Mathematical Programming, 91 (2002), 239.  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.  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.  doi: 10.1137/040603851.  Google Scholar

[11]

C. Helmberg, Semidefinite programming for combinatorial optimization,, Technical Report ZIB-Report ZR-00-34, (2000), 00.   Google Scholar

[12]

X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs,, Technical Report, (2003).   Google Scholar

[13]

F. Jarre, An interior method for nonconvex semidefinite programs,, Optimization and Engineering, 1 (2000), 347.  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.  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.  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, 29 (2007), 88.   Google Scholar

[17]

W. Sun, On filter methods for optimization,, The 3rd Australia-China Optimization Workshop, (2007).   Google Scholar

[18]

W. Sun, On filter-type methods for optimization: motivation and development,, An invited talk, (2008), 26.   Google Scholar

[19]

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

[20]

M. J. Todd, Semidefinite optimization,, Numerical Mathematics, 10 (2001), 515.   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).   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).   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.   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.   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.   Google Scholar

[1]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[2]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[3]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[4]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[5]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[6]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[7]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[8]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[9]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[11]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[12]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[13]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[15]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[16]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[17]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[19]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[20]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

 Impact Factor: 

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]