American Institute of Mathematical Sciences

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

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.
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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). Google Scholar [7] R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303. doi: 10.1137/S1052623402417298. Google Scholar [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. Google Scholar [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. Google Scholar [10] A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105. doi: 10.1007/PL00011370. Google Scholar [11] J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303. Google Scholar [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. Google Scholar [21] F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347. doi: 10.1023/A:1011562523132. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar [28] M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [39] M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283. Google Scholar [40] H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). Google Scholar [41] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [42] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268. doi: 10.1137/0312021. Google Scholar [43] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [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. Google Scholar [45] R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183. Google Scholar [46] A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301. doi: 10.1007/BF02614439. Google Scholar [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. Google Scholar [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. Google Scholar [49] D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [58] P. Tseng, Merit functions for semidefinite complementarity problems,, Math. Program., 83 (1998), 159. doi: 10.1007/BF02680556. Google Scholar [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. Google Scholar [60] Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented Lagrangian methods for semidefinite programming,, Optimization Online, (2009). Google Scholar [61] H. Yamashita, H. Yabe and K. Harada, A primal-dual interior point method for nonlinear semidefinite programming,, Technical report, (2007). Google Scholar [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. Google Scholar [63] S. Zhang, J. Ang and J. Sun, An alternating direction method for solving convex nonlinear semidefinite programming problems,, Technical Report, (2010). Google Scholar [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. Google Scholar

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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983). Google Scholar [7] R. Correa and C. H. Ramirez, A global algorithm for nonlinear semidefinite programming,, SIAM J. Optim., 15 (2004), 303. doi: 10.1137/S1052623402417298. Google Scholar [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. Google Scholar [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. Google Scholar [10] A. Forsgren, Optimality conditions for nonconvex semidefinite programming,, Math. Program., 88 (2000), 105. doi: 10.1007/PL00011370. Google Scholar [11] J. Eckstein, "Splitting Methods for Monotone Operators with Applications to Parallel Optimization,", PhD thesis, (1989). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. R. Hestenes, Multiplier and gradient methods,, J. Optim. Theory Appl., 4 (1969), 303. Google Scholar [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. Google Scholar [21] F. Jarre, An interior method for nonconvex semidefinite programs,, Optim. and Eng., 1 (2000), 347. doi: 10.1023/A:1011562523132. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [27] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar [28] M. Kocvara and M. Stingl, PENNON - A Generalized augmented Lagrangian method for semidefinite programming,, In, (2003), 297. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [39] M. J. D. Powell, A method for nonlinear constraints in minimization problems,, In, (1972), 283. Google Scholar [40] H. Qi and D. Sun, An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem,, IMA J. Numer. Anal., (2011). Google Scholar [41] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [42] R. T. Rockafellar, Augmented Lagrange multiplier functions and duality in nonconvex programming,, SIAM J. Control, 12 (1974), 268. doi: 10.1137/0312021. Google Scholar [43] R. T. Rockafellar, Monotone operators and the proximal point algorithm,, SIAM J. Control Optim., 14 (1976), 877. doi: 10.1137/0314056. Google Scholar [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. Google Scholar [45] R. T. Rockafellar, Lagrange multipliers and optimality,, SIAM Review, 35 (1993), 183. Google Scholar [46] A. Shapiro, First and second order analysis of nonlinear semidefinite programs,, Math. Program., 77 (1997), 301. doi: 10.1007/BF02614439. Google Scholar [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. Google Scholar [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. Google Scholar [49] D. Sun and J. Sun, Semismooth matrix valued functions,, Math. Oper. Res., 27 (2002), 150. doi: 10.1287/moor.27.1.150.342. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [57] N. K. Tsing, M. K. H. Fan and E. I. 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