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Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved
1. | School of Science, Dalian Nationalities University, Dalian, 116600, China |
2. | School of Mathematics, Liaoning Normal University, Dalian, 116029, China |
3. | College of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, China |
4. | School of Science, Dalian Ocean University, Dalian, 116023, China |
References:
[1] |
F. Alizadeh, Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM J. Optim., 5 (1995), 13-51.
doi: 10.1137/0805002. |
[2] |
P. Apkarian, D. Noll and H. D. Tuan, Fixed-order $H_\infty$ control design via a partially augmented lagrangian method, International Journal of Robust and Nonlinear Control, 13 (2003), 1137-1148.
doi: 10.1002/rnc.807. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[4] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.
doi: 10.1137/1.9781611970777. |
[5] |
C. Chen, T. C. Edwin Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, Journal of Industrial and Management Optimization, 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[6] |
M. Doljansky and M. Teboulle, An interior proximal algorithm and the exponential multiplier method for semidefinite programming, SIAM J. Optim., 9 (1999), 1-13.
doi: 10.1137/S1052623496309405. |
[7] |
B. Fares, P. Apkarian and D. Noll, An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, International Journal of Control, 74 (2001), 348-360.
doi: 10.1080/00207170010010605. |
[8] |
B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming, SIAM J. on Control and Optimization, 40 (2002), 1791-1820.
doi: 10.1137/S0363012900373483. |
[9] |
S. He and Y. Nie, A class of nonlinear Lagrangian algorithms for minimax problems, Journal of Industrial and Management Optimization, 9 (2013), 75-97.
doi: 10.3934/jimo.2013.9.75. |
[10] |
C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM J. Optim., 10 (2000), 673-696.
doi: 10.1137/S1052623497328987. |
[11] |
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511840371. |
[12] |
M. Kočvara and M. Stingl, Pennon: A code for convex nonlinear and semidefinite programming, Optimization Methods and Software, 18 (2003), 317-333.
doi: 10.1080/1055678031000098773. |
[13] |
Y. Li and L. Zhang, A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming, Journal of Industrial and Management Optimization, 5 (2009), 651-669.
doi: 10.3934/jimo.2009.5.651. |
[14] |
J. Lin, H. Chen and R. Sheu, Augmented Lagrange primal-dual approach for generalized fractional programming problems, Journal of Industrial and Management Optimization, 9 (2013), 723-741.
doi: 10.3934/jimo.2013.9.723. |
[15] |
L. Mosheyev and M. Zibulevsky, Penalty/Barrier multiplier algorithm for semidefinite programming, Optimization Methods and Software, 13 (2000), 235-261.
doi: 10.1080/10556780008805787. |
[16] |
D. Noll, Local convergence of an augmented Lagrangian method for matrix inequality constrained programming, Optimization Methods and Software, 22 (2007), 777-802.
doi: 10.1080/10556780701223970. |
[17] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods, Math. Programming Series B, 104 (2005), 701-727. |
[18] |
D. Noll, M. Torki and P. Apkarian, Partially augmented Lagrangian method for matrix inequality constraints, SIAM Journal on Optimization, 15 (2004), 161-184.
doi: 10.1137/S1052623402413963. |
[19] |
A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320.
doi: 10.1007/BF02614439. |
[20] |
A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization, Mathematics of Operations Research, 29 (2004), 479-491.
doi: 10.1287/moor.1040.0103. |
[21] |
M. Stingl, On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, Ph.D thesis, University of Erlangen, 2006. |
[22] |
D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391.
doi: 10.1007/s10107-007-0105-9. |
[23] |
M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.
doi: 10.1017/S0962492901000071. |
[24] |
L. Vanderberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95.
doi: 10.1137/1038003. |
[25] |
H. Wolkkowicz, R. Saigal and L. Vanderberghe, Handbook of Semidefinite Programming-Theory, Algorithms, and Applications, Kluwer Academic Publishers, 2000.
doi: 10.1007/978-1-4615-4381-7. |
[26] |
L. Zhang, Y. Li and J. Wu, Nonlinear rescaling Lagrangians for nonconvex semidefinite programming, Optimization, published online, 2013.
doi: 10.1080/02331934.2013.848861. |
[27] |
L. Zhang, J. Gu and X. Xiao, A class of nonlinear Lagrangians for nonconvex second order cone programming, Comput. Optim. Appl., 49 (2011), 61-99.
doi: 10.1007/s10589-009-9279-9. |
[28] |
L. Zhang, Y. Ren, Y. Wu and X. Xiao, A class of nonlinear Lagrangians: Theory and algorithm, Asia-Pacific Journal of Operational Research, 25 (2008), 327-371.
doi: 10.1142/S021759590800178X. |
[29] |
M. Zibulevski, Penalty Barrier Multiplier Methods for Large-Scale Nonlinear and Semidefinite Programming, Ph.D thesis, Technion, Israel, 1996. |
show all references
References:
[1] |
F. Alizadeh, Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM J. Optim., 5 (1995), 13-51.
doi: 10.1137/0805002. |
[2] |
P. Apkarian, D. Noll and H. D. Tuan, Fixed-order $H_\infty$ control design via a partially augmented lagrangian method, International Journal of Robust and Nonlinear Control, 13 (2003), 1137-1148.
doi: 10.1002/rnc.807. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[4] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.
doi: 10.1137/1.9781611970777. |
[5] |
C. Chen, T. C. Edwin Cheng, S. Li and X. Yang, Nonlinear augmented Lagrangian for nonconvex multiobjective optimization, Journal of Industrial and Management Optimization, 7 (2011), 157-174.
doi: 10.3934/jimo.2011.7.157. |
[6] |
M. Doljansky and M. Teboulle, An interior proximal algorithm and the exponential multiplier method for semidefinite programming, SIAM J. Optim., 9 (1999), 1-13.
doi: 10.1137/S1052623496309405. |
[7] |
B. Fares, P. Apkarian and D. Noll, An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, International Journal of Control, 74 (2001), 348-360.
doi: 10.1080/00207170010010605. |
[8] |
B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming, SIAM J. on Control and Optimization, 40 (2002), 1791-1820.
doi: 10.1137/S0363012900373483. |
[9] |
S. He and Y. Nie, A class of nonlinear Lagrangian algorithms for minimax problems, Journal of Industrial and Management Optimization, 9 (2013), 75-97.
doi: 10.3934/jimo.2013.9.75. |
[10] |
C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM J. Optim., 10 (2000), 673-696.
doi: 10.1137/S1052623497328987. |
[11] |
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511840371. |
[12] |
M. Kočvara and M. Stingl, Pennon: A code for convex nonlinear and semidefinite programming, Optimization Methods and Software, 18 (2003), 317-333.
doi: 10.1080/1055678031000098773. |
[13] |
Y. Li and L. Zhang, A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming, Journal of Industrial and Management Optimization, 5 (2009), 651-669.
doi: 10.3934/jimo.2009.5.651. |
[14] |
J. Lin, H. Chen and R. Sheu, Augmented Lagrange primal-dual approach for generalized fractional programming problems, Journal of Industrial and Management Optimization, 9 (2013), 723-741.
doi: 10.3934/jimo.2013.9.723. |
[15] |
L. Mosheyev and M. Zibulevsky, Penalty/Barrier multiplier algorithm for semidefinite programming, Optimization Methods and Software, 13 (2000), 235-261.
doi: 10.1080/10556780008805787. |
[16] |
D. Noll, Local convergence of an augmented Lagrangian method for matrix inequality constrained programming, Optimization Methods and Software, 22 (2007), 777-802.
doi: 10.1080/10556780701223970. |
[17] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods, Math. Programming Series B, 104 (2005), 701-727. |
[18] |
D. Noll, M. Torki and P. Apkarian, Partially augmented Lagrangian method for matrix inequality constraints, SIAM Journal on Optimization, 15 (2004), 161-184.
doi: 10.1137/S1052623402413963. |
[19] |
A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320.
doi: 10.1007/BF02614439. |
[20] |
A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization, Mathematics of Operations Research, 29 (2004), 479-491.
doi: 10.1287/moor.1040.0103. |
[21] |
M. Stingl, On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, Ph.D thesis, University of Erlangen, 2006. |
[22] |
D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391.
doi: 10.1007/s10107-007-0105-9. |
[23] |
M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.
doi: 10.1017/S0962492901000071. |
[24] |
L. Vanderberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95.
doi: 10.1137/1038003. |
[25] |
H. Wolkkowicz, R. Saigal and L. Vanderberghe, Handbook of Semidefinite Programming-Theory, Algorithms, and Applications, Kluwer Academic Publishers, 2000.
doi: 10.1007/978-1-4615-4381-7. |
[26] |
L. Zhang, Y. Li and J. Wu, Nonlinear rescaling Lagrangians for nonconvex semidefinite programming, Optimization, published online, 2013.
doi: 10.1080/02331934.2013.848861. |
[27] |
L. Zhang, J. Gu and X. Xiao, A class of nonlinear Lagrangians for nonconvex second order cone programming, Comput. Optim. Appl., 49 (2011), 61-99.
doi: 10.1007/s10589-009-9279-9. |
[28] |
L. Zhang, Y. Ren, Y. Wu and X. Xiao, A class of nonlinear Lagrangians: Theory and algorithm, Asia-Pacific Journal of Operational Research, 25 (2008), 327-371.
doi: 10.1142/S021759590800178X. |
[29] |
M. Zibulevski, Penalty Barrier Multiplier Methods for Large-Scale Nonlinear and Semidefinite Programming, Ph.D thesis, Technion, Israel, 1996. |
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