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January  2015, 11(1): 65-81. doi: 10.3934/jimo.2015.11.65

Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved

Yang Li 1, , Yonghong Ren 2, , Yun Wang 3, and  Jian Gu 4,

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

Received  August 2012 Revised  December 2013 Published  May 2014

In this paper, we analyze the convergence properties of a nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming (NCSDP) problems. It is different from other convergence analysis, because the subproblem in our algorithm is inexactly solved. Under the constraint nondegeneracy condition, the strict complementarity condition and the second order sufficient conditions, it is obtained that the nonlinear Lagrangian algorithm proposed is locally convergent by choosing a proper stopping criterion and the error bound of solution is proportional to the penalty parameter when the penalty parameter is less than a threshold.
Keywords: nonconvex semidefinite programming, Löwner operator, nonlinear Lagrangian method, Log-Sigmoid function., Inexact solution.
Mathematics Subject Classification: Primary: 90C22, 90C26; Secondary: 90C3.
Citation: Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial & Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM J. Optim., 10 (2000), 673-696. doi: 10.1137/S1052623497328987.  Google Scholar

[11]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. doi: 10.1017/CBO9780511840371.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439.  Google Scholar

[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.  Google Scholar

[21]

M. Stingl, On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, Ph.D thesis, University of Erlangen, 2006. Google Scholar

[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.  Google Scholar

[23]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560. doi: 10.1017/S0962492901000071.  Google Scholar

[24]

L. Vanderberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Zibulevski, Penalty Barrier Multiplier Methods for Large-Scale Nonlinear and Semidefinite Programming, Ph.D thesis, Technion, Israel, 1996. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM J. Optim., 10 (2000), 673-696. doi: 10.1137/S1052623497328987.  Google Scholar

[11]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991. doi: 10.1017/CBO9780511840371.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[19]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439.  Google Scholar

[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.  Google Scholar

[21]

M. Stingl, On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, Ph.D thesis, University of Erlangen, 2006. Google Scholar

[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.  Google Scholar

[23]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560. doi: 10.1017/S0962492901000071.  Google Scholar

[24]

L. Vanderberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), 49-95. doi: 10.1137/1038003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Zibulevski, Penalty Barrier Multiplier Methods for Large-Scale Nonlinear and Semidefinite Programming, Ph.D thesis, Technion, Israel, 1996. Google Scholar

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