On methods for solving nonlinear semidefinite optimization problems

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2011, 1(1): 1-14. doi: 10.3934/naco.2011.1.1

On methods for solving nonlinear semidefinite optimization problems

1.	School of Business, National University of Singapore, 119245, Singapore

Received November 2010 Revised November 2010 Published February 2011

Abstract
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The nonlinear semidefinite optimization problem arises from applications in system control, structural design, financial management, and other fields. However, much work is yet to be done to effectively solve this problem. We introduce some new theoretical and algorithmic development in this field. In particular, we discuss first and second-order algorithms that appear to be promising, which include the alternating direction method, the augmented Lagrangian method, and the smoothing Newton method. Convergence theorems are presented and preliminary numerical results are reported.

Keywords: Alternating direction method, augmented Lagrangian method, semismooth Newton method..

Mathematics Subject Classification: Primary: 90C22; Secondary: 90C2.

Citation: Jie Sun. On methods for solving nonlinear semidefinite optimization problems. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 1-14. doi: 10.3934/naco.2011.1.1

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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