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Recent advances in numerical methods for nonlinear equations and nonlinear least squares
On methods for solving nonlinear semidefinite optimization problems
1. | School of Business, National University of Singapore, 119245, Singapore |
References:
[1] |
A. Ben-Tal, F. Jarre, M. Kocvara, A. Nemirovski and J. Zowe, Optimal design of trusses under a nonconvex global buckling constraint,, Optim. and Eng., 1 (2000), 189.
doi: 10.1023/A:1010091831812. |
[2] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM Studies in Applied Mathematics, (1994).
|
[3] |
X. Chen, H. D. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems,, SIAM J. Optim., 13 (2003), 960.
doi: 10.1137/S1052623400380584. |
[4] |
X. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, , Comput. Optim. Appl., 25 (2003), 39.
doi: 10.1023/A:1022996819381. |
[5] |
X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems,, Math. Program., 95 (2003), 431.
doi: 10.1007/s10107-002-0306-1. |
[6] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).
|
[7] |
R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303.
doi: 10.1137/S1052623402417298. |
[8] |
M. Diehl, F. Jarre and C. H. Vogelbusch, Loss of superlinear convergence for an SQP-type method with conic constraints,, SIAM J. Optim., 16 (2006), 1201.
doi: 10.1137/050625977. |
[9] |
M. Doljansky, An interior proximal algorithm and the exponential multiplier method for semidefinite programming,, SIAM J. Optim., 9 (1999), 1.
doi: 10.1137/S1052623496309405. |
[10] |
A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105.
doi: 10.1007/PL00011370. |
[11] |
J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). |
[12] |
B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM J. Contr. and Optim., 40 (2002), 1791.
doi: 10.1137/S0363012900373483. |
[13] |
M. L. Flegel and C. Kanzow, Equivalence of two nondegeneracy conditions for semidefinite programs,, J. Optim. Theory Appl., 135 (2007), 381.
doi: 10.1007/s10957-007-9270-5. |
[14] |
M. Fukushima, Application of the alternating directions method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[15] |
M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM J. Optim., 12 (2002), 436.
doi: 10.1137/S1052623400380365. |
[16] |
Y. Gao and D. Sun, Calibrating least squares covariance matrix problems with equality and inequality constraints,, SIAM J. Matrix Anal. Appl. 31 (2009), 31 (2009), 1432.
doi: 10.1137/080727075. |
[17] |
B. S. He, L. Z. Liao, D. R. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
|
[18] |
B. S. He, L. Z. Liao and M. J. Qian, Alternating projection based prediction-correction methods for structured variational inequalities,, J. Compu. Math., 24 (2002), 693.
|
[19] |
M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303.
|
[20] |
N. J. Higham, Computing a nearest symmetric positive semidefinite matrix,, Linear Algebra Appl., 103 (1998), 103.
doi: 10.1016/0024-3795(88)90223-6. |
[21] |
F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347.
doi: 10.1023/A:1011562523132. |
[22] |
C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM J. Optim., 20 (2009), 297.
doi: 10.1137/060657662. |
[23] |
C. Kanzow and C. Nagel, Semidefinite programs: New search directions, smoothing-type methods, and numerical results,, SIAM J. Optim., 13 (2002), 1.
doi: 10.1137/S1052623401390525. |
[24] |
C. Kanzow and C. Nagel, Some structural properties of a Newton-type method for semidefinite programs,, J. Optim. Theory Appl., 122 (2004), 219.
doi: 10.1023/B:JOTA.0000041737.19689.4c. |
[25] |
C. Kanzow and C. Nagel, Quadratic convergence of a nonsmooth newton-type method for semidefinite programs without strict complementarity,, SIAM J. Optim., 15 (2005), 654.
|
[26] |
C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Successive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251.
doi: 10.1007/s10589-005-3231-4. |
[27] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).
|
[28] |
M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. |
[29] |
M. Kocvara and M. Stingl, PENNON: a code for convex nonlinear and semidefinite programming,, Optim. Meth. Soft., 18 (2003), 317.
doi: 10.1080/1055678031000098773. |
[30] |
M. Kocvara and M. Stingl, Solving nonconvex SDP problems of structural optimization with stability control,, Optim. Meth. Soft., 19 (2004), 595.
doi: 10.1080/10556780410001682844. |
[31] |
L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM J. Optim., 19 (2008), 1028.
doi: 10.1137/060676775. |
[32] |
F. Leibfritz, COMP $ l_{ e}$ ib 1.1: Constraint matrix-optimization problem library - a collection of test examples for nonlinear semidefinite programs, control system design and related problems,, Technical Report, (2005). |
[33] |
F. Leibfritz and M. E. Mostafa, An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems,, SIAM J. Optim., 12 (2002), 1048.
doi: 10.1137/S1052623400375865. |
[34] |
C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, China Sci. Ser. A, 52 (2009), 2341.
doi: 10.1007/s11425-009-0168-6. |
[35] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods,, Math. Program., 104 (2005), 701.
doi: 10.1007/s10107-005-0634-z. |
[36] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods,, Math. Program., 104 (2005), 729.
doi: 10.1007/s10107-005-0635-y. |
[37] |
J. S. Pang, D. Sun and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems,, Math. Oper. Res., 28 (2003), 39.
doi: 10.1287/moor.28.1.39.14258. |
[38] |
T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity,, Math. Oper. Res., 27 (2002), 170.
doi: 10.1287/moor.27.1.170.331. |
[39] |
M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283.
|
[40] |
H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). |
[41] |
L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353.
doi: 10.1007/BF01581275. |
[42] |
R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268.
doi: 10.1137/0312021. |
[43] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877.
doi: 10.1137/0314056. |
[44] |
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97.
doi: 10.1287/moor.1.2.97. |
[45] |
R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183.
|
[46] |
A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301.
doi: 10.1007/BF02614439. |
[47] |
A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization,, Math. Oper. Res., 29 (2004), 479.
doi: 10.1287/moor.1040.0103. |
[48] |
D. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Math. Oper. Res., 31 (2006), 761.
doi: 10.1287/moor.1060.0195. |
[49] |
D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150.
doi: 10.1287/moor.27.1.150.342. |
[50] |
D. Sun and J. Sun, Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions,, Math. Program., 103 (2005), 575.
doi: 10.1007/s10107-005-0577-4. |
[51] |
D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Math. Oper. Res., 33 (2008), 421.
doi: 10.1287/moor.1070.0300. |
[52] |
D. Sun, J. Sun and L. Zhang, Rates of convergence of the augmented Lagrangian method for nonlinear semidefinite programming,, Math. Program., 114 (2008), 349.
doi: 10.1007/s10107-007-0105-9. |
[53] |
J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM J. Optim., 14 (2004), 783.
doi: 10.1137/S1052623400379620. |
[54] |
J. Sun, L. Zhang and Y. Wu, Properties of the augmented Lagrangian in nonlinear semidefinite optimization,, J. Optim. Theory Appl., 129 (2006), 437.
doi: 10.1007/s10957-006-9078-8. |
[55] |
J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, Eur. J. Oper. Res., 207 (2010), 1210.
doi: 10.1016/j.ejor.2010.07.020. |
[56] |
J. Sun, G. Zhao and J. Zhu, A predictor-corrector algorithm for a class of nonlinear saddle point problems,, SIAM J. Contr. Optim., 35 (1997), 532.
doi: 10.1137/S0363012994276111. |
[57] |
N. K. Tsing, M. K. H. Fan and E. I. Verriest, On analyticity of functions involving eigenvalues,, Linear Algebra Appl., 207 (1994), 159.
doi: 10.1016/0024-3795(94)90009-4. |
[58] |
P. Tseng, Merit functions for semidefinite complementarity problems,, Math. Program., 83 (1998), 159.
doi: 10.1007/BF02680556. |
[59] |
C. Wang, D. Sun and K. C. Toh, Solving log-determinant optimization problems by a Newton-CG proximal point algorithm,, SIAM J. Optim., 20 (2010), 2994.
doi: 10.1137/090772514. |
[60] |
Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming,, Optimization Online, (2009). |
[61] |
H. Yamashita, H. Yabe and K. Harada, A primal-dual interior point method for nonlinear semidefinite programming,, Technical report, (2007). |
[62] |
Z. S. Yu, Solving semidefinite programming problems via alternating direction methods,, J. Compu. Appl. Math., 193 (2006), 437.
doi: 10.1016/j.cam.2005.07.002. |
[63] |
S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Technical Report, (2010). |
[64] |
X. Zhao, D. Sun and K. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming,, SIAM J. Optim., 20 (2010), 1737.
doi: 10.1137/080718206. |
show all references
References:
[1] |
A. Ben-Tal, F. Jarre, M. Kocvara, A. Nemirovski and J. Zowe, Optimal design of trusses under a nonconvex global buckling constraint,, Optim. and Eng., 1 (2000), 189.
doi: 10.1023/A:1010091831812. |
[2] |
S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory,", SIAM Studies in Applied Mathematics, (1994).
|
[3] |
X. Chen, H. D. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems,, SIAM J. Optim., 13 (2003), 960.
doi: 10.1137/S1052623400380584. |
[4] |
X. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, , Comput. Optim. Appl., 25 (2003), 39.
doi: 10.1023/A:1022996819381. |
[5] |
X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems,, Math. Program., 95 (2003), 431.
doi: 10.1007/s10107-002-0306-1. |
[6] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).
|
[7] |
R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303.
doi: 10.1137/S1052623402417298. |
[8] |
M. Diehl, F. Jarre and C. H. Vogelbusch, Loss of superlinear convergence for an SQP-type method with conic constraints,, SIAM J. Optim., 16 (2006), 1201.
doi: 10.1137/050625977. |
[9] |
M. Doljansky, An interior proximal algorithm and the exponential multiplier method for semidefinite programming,, SIAM J. Optim., 9 (1999), 1.
doi: 10.1137/S1052623496309405. |
[10] |
A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105.
doi: 10.1007/PL00011370. |
[11] |
J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). |
[12] |
B. Fares, D. Noll and P. Apkarian, Robust control via sequential semidefinite programming,, SIAM J. Contr. and Optim., 40 (2002), 1791.
doi: 10.1137/S0363012900373483. |
[13] |
M. L. Flegel and C. Kanzow, Equivalence of two nondegeneracy conditions for semidefinite programs,, J. Optim. Theory Appl., 135 (2007), 381.
doi: 10.1007/s10957-007-9270-5. |
[14] |
M. Fukushima, Application of the alternating directions method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[15] |
M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM J. Optim., 12 (2002), 436.
doi: 10.1137/S1052623400380365. |
[16] |
Y. Gao and D. Sun, Calibrating least squares covariance matrix problems with equality and inequality constraints,, SIAM J. Matrix Anal. Appl. 31 (2009), 31 (2009), 1432.
doi: 10.1137/080727075. |
[17] |
B. S. He, L. Z. Liao, D. R. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
|
[18] |
B. S. He, L. Z. Liao and M. J. Qian, Alternating projection based prediction-correction methods for structured variational inequalities,, J. Compu. Math., 24 (2002), 693.
|
[19] |
M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303.
|
[20] |
N. J. Higham, Computing a nearest symmetric positive semidefinite matrix,, Linear Algebra Appl., 103 (1998), 103.
doi: 10.1016/0024-3795(88)90223-6. |
[21] |
F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347.
doi: 10.1023/A:1011562523132. |
[22] |
C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM J. Optim., 20 (2009), 297.
doi: 10.1137/060657662. |
[23] |
C. Kanzow and C. Nagel, Semidefinite programs: New search directions, smoothing-type methods, and numerical results,, SIAM J. Optim., 13 (2002), 1.
doi: 10.1137/S1052623401390525. |
[24] |
C. Kanzow and C. Nagel, Some structural properties of a Newton-type method for semidefinite programs,, J. Optim. Theory Appl., 122 (2004), 219.
doi: 10.1023/B:JOTA.0000041737.19689.4c. |
[25] |
C. Kanzow and C. Nagel, Quadratic convergence of a nonsmooth newton-type method for semidefinite programs without strict complementarity,, SIAM J. Optim., 15 (2005), 654.
|
[26] |
C. Kanzow, C. Nagel, H. Kato and M. Fukushima, Successive linearization methods for nonlinear semidefinite programs,, Comput. Optim. Appl., 31 (2005), 251.
doi: 10.1007/s10589-005-3231-4. |
[27] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).
|
[28] |
M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. |
[29] |
M. Kocvara and M. Stingl, PENNON: a code for convex nonlinear and semidefinite programming,, Optim. Meth. Soft., 18 (2003), 317.
doi: 10.1080/1055678031000098773. |
[30] |
M. Kocvara and M. Stingl, Solving nonconvex SDP problems of structural optimization with stability control,, Optim. Meth. Soft., 19 (2004), 595.
doi: 10.1080/10556780410001682844. |
[31] |
L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM J. Optim., 19 (2008), 1028.
doi: 10.1137/060676775. |
[32] |
F. Leibfritz, COMP $ l_{ e}$ ib 1.1: Constraint matrix-optimization problem library - a collection of test examples for nonlinear semidefinite programs, control system design and related problems,, Technical Report, (2005). |
[33] |
F. Leibfritz and M. E. Mostafa, An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems,, SIAM J. Optim., 12 (2002), 1048.
doi: 10.1137/S1052623400375865. |
[34] |
C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programming,, China Sci. Ser. A, 52 (2009), 2341.
doi: 10.1007/s11425-009-0168-6. |
[35] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods,, Math. Program., 104 (2005), 701.
doi: 10.1007/s10107-005-0634-z. |
[36] |
D. Noll and P. Apkarian, Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods,, Math. Program., 104 (2005), 729.
doi: 10.1007/s10107-005-0635-y. |
[37] |
J. S. Pang, D. Sun and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems,, Math. Oper. Res., 28 (2003), 39.
doi: 10.1287/moor.28.1.39.14258. |
[38] |
T. Pennanen, Local convergence of the proximal point algorithm and multiplier methods without monotonicity,, Math. Oper. Res., 27 (2002), 170.
doi: 10.1287/moor.27.1.170.331. |
[39] |
M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283.
|
[40] |
H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). |
[41] |
L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353.
doi: 10.1007/BF01581275. |
[42] |
R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268.
doi: 10.1137/0312021. |
[43] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877.
doi: 10.1137/0314056. |
[44] |
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming,, Math. Oper. Res., 1 (1976), 97.
doi: 10.1287/moor.1.2.97. |
[45] |
R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183.
|
[46] |
A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301.
doi: 10.1007/BF02614439. |
[47] |
A. Shapiro and J. Sun, Some properties of the augmented Lagrangian in cone constrained optimization,, Math. Oper. Res., 29 (2004), 479.
doi: 10.1287/moor.1040.0103. |
[48] |
D. Sun, The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications,, Math. Oper. Res., 31 (2006), 761.
doi: 10.1287/moor.1060.0195. |
[49] |
D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150.
doi: 10.1287/moor.27.1.150.342. |
[50] |
D. Sun and J. Sun, Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions,, Math. Program., 103 (2005), 575.
doi: 10.1007/s10107-005-0577-4. |
[51] |
D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Math. Oper. Res., 33 (2008), 421.
doi: 10.1287/moor.1070.0300. |
[52] |
D. Sun, J. Sun and L. Zhang, Rates of convergence of the augmented Lagrangian method for nonlinear semidefinite programming,, Math. Program., 114 (2008), 349.
doi: 10.1007/s10107-007-0105-9. |
[53] |
J. Sun, D. Sun and L. Qi, A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems,, SIAM J. Optim., 14 (2004), 783.
doi: 10.1137/S1052623400379620. |
[54] |
J. Sun, L. Zhang and Y. Wu, Properties of the augmented Lagrangian in nonlinear semidefinite optimization,, J. Optim. Theory Appl., 129 (2006), 437.
doi: 10.1007/s10957-006-9078-8. |
[55] |
J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, Eur. J. Oper. Res., 207 (2010), 1210.
doi: 10.1016/j.ejor.2010.07.020. |
[56] |
J. Sun, G. Zhao and J. Zhu, A predictor-corrector algorithm for a class of nonlinear saddle point problems,, SIAM J. Contr. Optim., 35 (1997), 532.
doi: 10.1137/S0363012994276111. |
[57] |
N. K. Tsing, M. K. H. Fan and E. I. Verriest, On analyticity of functions involving eigenvalues,, Linear Algebra Appl., 207 (1994), 159.
doi: 10.1016/0024-3795(94)90009-4. |
[58] |
P. Tseng, Merit functions for semidefinite complementarity problems,, Math. Program., 83 (1998), 159.
doi: 10.1007/BF02680556. |
[59] |
C. Wang, D. Sun and K. C. Toh, Solving log-determinant optimization problems by a Newton-CG proximal point algorithm,, SIAM J. Optim., 20 (2010), 2994.
doi: 10.1137/090772514. |
[60] |
Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming,, Optimization Online, (2009). |
[61] |
H. Yamashita, H. Yabe and K. Harada, A primal-dual interior point method for nonlinear semidefinite programming,, Technical report, (2007). |
[62] |
Z. S. Yu, Solving semidefinite programming problems via alternating direction methods,, J. Compu. Appl. Math., 193 (2006), 437.
doi: 10.1016/j.cam.2005.07.002. |
[63] |
S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Technical Report, (2010). |
[64] |
X. Zhao, D. Sun and K. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming,, SIAM J. Optim., 20 (2010), 1737.
doi: 10.1137/080718206. |
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