[1] Anderson T W. Estimating linear restrictions on regression coefficients for multivariate normal distributions[J]. Annals of Mathematical Statistics, 1951, 22(3):327-351.
[2] El Ghaoui L, Gahinet P. Rank minimization under LMI constraints:A framework for output feedback problems[C]//Proceedings European Control Conference, 1993.
[3] Fazel M. Matrix Rank Minimization with Applications[D]. Stanford:Stanford University, 2002.
[4] Natarajan B. Sparse approximate solutions to linear systems[J]. SIAM Journal on Computing, 1995, 24(2):227-234.
[5] David J. Algorithms for Analysis and Design of Robust Controllers[D]. Leuven:University of Leuven, 1994.
[6] Delgado R A, Agüero J C, Goodwin G C. A rank-constrained optimization approach:Application to factor analysis[J]. IFAC Proceedings Volumes, 2014, 47(3):10373-10378.
[7] Chen Y, Wainwright M J. Fast low-rank estimation by projected gradient descent:General statistical and algorithmic guarantees[J]. arXiv:1509.03025.
[8] She Y, Chen K. Robust reduced-rank regression[J]. Biometrika, 2017, 104(3):633-647.
[9] Delvaux S, Van Barel M. A Givens-weight representation for rank structured matrices[J]. SIAM Journal on Matrix Analysis and Applications, 2007, 29(4):1147-1170.
[10] Wright J, Ma Y, Mairal J, et al. Sparse representation for computer vision and pattern recognition[J]. Proceedings of the IEEE, 2010, 98(6):1031-1044.
[11] Xue S, Jin X. Robust classwise and projective low-rank representation for image classification[J]. Signal, Image and Video Processing, 2018, 12(1):107-115.
[12] Xing E P, Jordan M I, Russell S J, et al. Distance metric learning with application to clustering with side-information[C]//International Conference on Neural Information Processing Systems, 2002, 521-528.
[13] Amit Y, Fink M, Srebro N, et al. Uncovering shared structures in multiclass classification[C]//Proceedings of the 24th International Conference on Machine Learning, 2007, 17-24.
[14] Gillis N, Glineur F. Low-rank matrix approximation with weights or missing data is NP-hard[J]. SIAM Journal on Matrix Analysis and Applications, 2011, 32(4):1149-1165.
[15] Kobayashi T. Low-rank bilinear classification:Efficient convex optimization and extensions[J]. International Journal of Computer Vision, 2014, 110(3):308-327.
[16] Zeng W J, So H C. Outlier-robust matrix completion via lp-minimization[J]. IEEE Transactions on Signal Processing, 2017, 66(5):1125-1140.
[17] Shechtman Y, Eldar Y C, Cohen O, et al. Phase retrieval with application to optical imaging:A contemporary overview[J]. IEEE Signal Processing Magazine, 2015, 32(3):87-109.
[18] Ahmed A, Romberg J. Compressive multiplexing of correlated signals[J]. IEEE Transactions on Information Theory, 2014, 61(1):479-498.
[19] Fallat S M, Hogben L. The minimum rank of symmetric matrices described by a graph:A survey[J]. Linear Algebra and its Applications, 2007, 426(2-3):558-582.
[20] Ames B P W, Vavasis S A. Nuclear norm minimization for the planted clique and biclique problems[J]. Mathematical Programming, 2009, 129(1):69-89.
[21] Dong W, Shi G, Li X, et al. Compressive sensing via nonlocal low-rank regularization[J]. IEEE Transactions on Image Processing, 2014, 23(8):3618-3632.
[22] Argyriou A, Evgeniou T, Pontil M. Multi-task feature learning[C]//Advances in Neural Information Processing Systems, 2007, 41-48.
[23] Cheng B, Liu G, Wang J, et al. Multi-task low-rank affinity pursuit for image segmentation[C]//2011 International Conference on Computer Vision, 2011, 2439-2446.
[24] Gross D, Liu Y K, Flammia S T. Quantum state tomography via compressed sensing[J]. Physical Review Letters, 2010, 105(15):150401.
[25] Arslan G, Demirkol M F, Song Y. Equilibrium efficiency improvement in MIMO interference systems:A decentralized stream control approach[J]. IEEE Transactions on Wireless Communications, 2007, 6(8):2984-2993.
[26] Flammia S T, Gross D, Liu Y K, et al. Quantum tomography via compressed sensing:Error bounds, sample complexity and efficient estimators[J]. New Journal of Physics, 2012, 14(9):095022.
[27] Kalev A, Kosut R L, Deutsch I H. Quantum tomography protocols with positivity are compressed sensing protocols[J]. NPJ Quantum Information, 2015, 1(1):1-6.
[28] Wang B, Hu Y, Gao J, et al. Localized LRR on Grassmann manifold:An extrinsic view[J]. IEEE Transactions on Circuits and Systems for Video Technology, 2017, 28(10):2524-2536.
[29] Orsi R, Helmke U, Moore J B. A Newton-like method for solving rank constrained linear matrix inequalities[J]. Automatica, 2006, 42(11):1875-1882.
[30] Mesbahi M, Papavassilopoulos G P. On the rank minimization problem over a positive semidefinite linear matrix inequality[J]. IEEE Transactions on Automatic Control, 1997, 42(2):239-243.
[31] Fazel M, Hindi H, Boyd S P. A rank minimization heuristic with application to minimum order system approximation[C]//Proceedings of the 2001 American Control Conference, 2001, 6:4734-4739.
[32] Fazel M, Pong T K, Sun D, et al. Hankel matrix rank minimization with applications to system identification and realization[J]. SIAM Journal on Matrix Analysis and Applications, 2013, 34(3):946-977.
[33] Ankelhed D. On Design of Low Order H-Infinity Controllers[D]. Linkoping:Linkoping University, 2011.
[34] Kim S J, Moon Y H. Structurally constrained H2 and H∞ control:A rank-constrained LMI approach[J]. Automatica, 2006, 42(9):1583-1588.
[35] Zare A, Chen Y, Jovanovic M R, et al. Low-complexity modeling of partially available secondorder statistics:Theory and an efficient matrix completion algorithm[J]. IEEE Transactions on Automatic Control, 2016, 62(3):1368-1383.
[36] Pietersz R, Groenen P J F. Rank reduction of correlation matrices by majorization[J]. Quantitative Finance, 2004, 4(6):649-662.
[37] Brigo D. Interest rate models-theory and practice:With smile, inflation and credit[J]. Springer Finance, 2006, 11(4):559-572.
[38] Rebonato R. Modern Pricing of Interest-Rate Derivatives:The LIBOR Market Model and Beyond[M]. Princeton:Princeton University Press, 2012.
[39] Wu L. Fast at-the-money calibration of the LIBOR market model through Lagrange multipliers[J]. Journal of Computational Finance, 2002, 6(2):39-77.
[40] Zhang Z, Wu L. Optimal low-rank approximation to a correlation matrix[J]. Linear Algebra and its Applications, 2003, 364(1):161-187.
[41] Lillo F, Mantegna R. Spectral density of the correlation matrix of factor models:A random matrix theory approach[J]. Physical Review E, 2005, 72(1):016219.
[42] Negahban S, Wainwright M J. Estimation of (near) low-rank matrices with noise and high dimensional scaling[J]. Annals of Statistics, 2011, 39(2):1069-1097.
[43] Bach F R. Consistency of trace norm minimization[J]. Journal of Machine Learning Research, 2008, 9(Jun):1019-1048.
[44] Bunea F, She Y, Wegkamp M H. Joint variable and rank selection for parsimonious estimation of high-dimensional matrices[J]. Annals of Statistics, 2012, 40(5):2359-2388.
[45] Zhou H, Li L. Regularized matrix regression. Journal of the Royal Statistical Society[J]. 2014, 76(2):463-483.
[46] Wainwright M J. Structured regularizers for high-dimensional problems:Statistical and computational issues[J]. Annual Review of Statistics and its Application, 2014, 1(1):233-253.
[47] Bing X, Wegkamp M H. Adaptive estimation of the rank of the coefficient matrix in high dimensional multivariate response regression models[J]. Annals of Statistics, 2019, 47(6):3157-3184.
[48] Wright J, Ganesh A, Rao S, et al. Robust principal component analysis:Exact recovery of corrupted low-rank matrices via convex optimization[C]//Advances in Neural Information Processing Systems, 2009, 2080-2088.
[49] Lin Z, Ganesh A, Wright J, et al. Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix[C]//Journal of the Marine Biological Association of the United Kingdom, 2009, 56:707-722.
[50] Candès E J, Li X, Ma Y, et al. Robust principal component analysis[J]. Journal of the ACM, 2011, 58(3):1-37.
[51] Chandrasekaran V, Sanghavi S, Parrilo P A, et al. Sparse and low-rank matrix decompositions[J]. IFAC Proceedings Volumes, 2009, 42(10):1493-1498.
[52] Chandrasekaran V, Sanghavi S, Parrilo P A, et al. Rank-sparsity incoherence for matrix decomposition[J]. SIAM Journal on Optimization, 2011, 21(2):572-596.
[53] Berge J, Kiers H. A numerical approach to the approximate and the exact minimum rank of a covariance matrix[J]. Psychometrika, 1991, 56(2):309-315.
[54] Bertsimas D, Copenhaver M S, Mazumder R. Certifiably optimal low rank factor analysis[J]. Journal of Machine Learning Research, 2016, 18(29):1-53.
[55] Khamaru K, Mazumder R. Computation of the maximum likelihood estimator in low-rank factor analysis[J]. Mathematical Programming, 2019, 176(1-2):279-310.
[56] Silverman L. Realization of linear dynamical systems[J]. IEEE Transactions on Automatic Control, 1971, 16(6):554-567.
[57] Grussler C, Rantzer A, Giselsson P. Low-rank optimization with convex constraints[J]. IEEE Transactions on Automatic Control, 2018, 63(11):4000-4007.
[58] Qi H, Shen J, Xiu N. A sequential majorization method for approximating weighted time series of finite rank[J]. Statistics and its Interface, 2018, 11(4):615-630.
[59] Xing E P, Jordan M I, Russell S J, et al. Distance metric learning with application to clustering with side-information[C]//International Conference on Neural Information Processing Systems, 2002, 521-528.
[60] Weinberger K Q, Saul L K. Distance metric learning for large margin nearest neighbor classification[J]. Journal of Machine Learning Research, 2009, 10(Feb):207-244.
[61] Davis J V, Kulis B, Jain P, et al. Information-theoretic metric learning[C]//Proceedings of the 24th International Conference on Machine Learning, 2007, 209-216.
[62] Luo L, Xie Y, Zhang Z, et al. Support matrix machines[C]//In International Conference on Machine Learning, 2015, 938-947.
[63] Qian C, Tran-Dinh Q, Fu S, et al. Robust multicategory support matrix machines[J]. Mathematical Programming, 2019, 176(1-2):429-463.
[64] Razzak I, Blumenstein M, Xu G. Multiclass support matrix machines by maximizing the interclass margin for single trial EGG classification[J]. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 2019, 27(6):1117-1127.
[65] Duan B, Yuan J, Liu Y, et al. Quantum algorithm for support matrix machines[J]. Physical Review A, 2017, 96(3):032301.
[66] Markovsky I. Low Rank Approximation[M]. London:Springer, 2012.
[67] Davenport M A, Romberg J. An overview of low-rank matrix recovery from incomplete observations[J]. IEEE Journal of Selected Topics in Signal Processing, 2016, 10(4):608-622.
[68] Gao Y. Structured Low Rank Matrix Optimization Problems:A Penalty Approach[D]. Singapore:National University of Singapore, 2010.
[69] Ding C. An Introduction to a Class of Matrix Optimization Problems[D]. Singapore:National University of Singapore, 2012.
[70] Miao W. Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints[D]. Singapore:National University of Singapore, 2013.
[71] Bi S. Study for Multi-Stage Convex Relaxation Approach to Low-Rank Optimization Problems[D]. Guangzhou:South China University of Technology, 2014.
[72] Zheng F. Study on Models and Algorithms for Matrix Rank Minimization Problem[D]. Wuhan:Wuhan University, 2015.
[73] Chen Y. Algorithms for Low-Rank Semidefinite Matrix Recovery Problems[D]. Beijing:Beijing Jiaotong University, 2015.
[74] Recht B, Fazel M, Parrilo P A. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization[J]. SIAM Review, 2010, 52(3):471-501.
[75] Recht B, Xu W, Hassibi B. Null space conditions and thresholds for rank minimization[J]. Mathematical Programming, 2011, 127(1):175-202.
[76] Kong L, Sun J, Xiu N. S-semigoodness for low-rank semidefinite matrix recovery[J]. Pacific Journal of Optimization, 2014, 10(1):73-83.
[77] Cai J F, Candes E J, Shen Z. A singular value thresholding algorithm for matrix completion[J]. SIAM Journal on Optimization, 2010, 20(4):1956-1982.
[78] Liu Z, Vandenberghe L. Interior-point method for nuclear norm approximation with application to system identification[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31(3):1235-1256.
[79] Ma S, Goldfarb D, Chen L. Fixed point and Bregman iterative methods for matrix rank minimization[J]. Mathematical Programming, 2011, 128(1-2):321-353.
[80] Toh K C, Yun S. An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems[J]. Pacific Journal of Optimization, 2010, 6(3):615-640.
[81] Yang J, Yuan X. Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization[J]. Mathematics of Computation, 2013, 82(281):301-329.
[82] Huang B, Ma S, Goldfarb D. Accelerated linearized Bregman method[J]. Journal of Scientific Computing, 2013, 54(2-3):428-453.
[83] Liu Y J, Sun D, Toh K C. An implementable proximal point algorithmic framework for nuclear norm minimization[J]. Mathematical Programming, 2012, 133(1-2):399-436.
[84] Marjanovic G, Solo V. On lq optimization and matrix completion[J]. IEEE Transactions on Signal Processing, 2010, 60(11):5714-5724.
[85] Nie F, Wang H, Cai X, et al. Robust matrix completion via joint Schatten p-norm and lp-norm minimization[C]//IEEE International Conference on Data Mining, 2012, 566-574.
[86] Mohan K, Fazel M. Iterative reweighted algorithms for matrix rank minimization[J]. Journal of Machine Learning Research, 2012, 13(Nov):3441-3473.
[87] Chen Y, Xiu N, Peng D. Global solutions of non-Lipschitz s2-sp minimization over the positive semidefinite cone[J]. Optimization Letters, 2014, 8(7):2053-2064.
[88] Lu Z, Yong Z, Jian L. lp regularized low-rank approximation via iterative reweighted singular value minimization[J]. Computational Optimization and Applications, 2017, 68(3):1-24.
[89] Kong L, Xiu N. Exact low-rank matrix recovery via nonconvex Schatten p-minimization[J]. Asia-Pacific Journal of Operational Research, 2013, 30(3):1340010.
[90] Oymak S, Mohan K, Fazel M, et al. A simplified approach to recovery conditions for low rank matrices[C]//2011 IEEE International Symposium on Information Theory Proceedings, 2011, 2318-2322.
[91] Fornasier M, Rauhut H, Ward R. Low-rank matrix recovery via iteratively reweighted least squares minimization[J]. SIAM Journal on Optimization, 2011, 21(4):1614-1640.
[92] Lu Y, Zhang L, Wu J. A smoothing majorization method for matrix minimization[J]. Optimization Methods and Software, 2015, 30(4):682-705.
[93] Peng D, Xiu N, Yu J. s1/2 regularization methods and fixed point algorithms for affine rank minimization problems[J]. Computational Optimization and Applications, 2017, 67(3):543-569.
[94] Fazel M, Hindi H, Boyd S P. Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices[J]. American Control Conference, 2003, 3:2156-2162.
[95] Mohan K, Fazel M. New restricted isometry results for noisy low-rank recovery[C]//IEEE International Symposium on Information Theory, 2010, 1573-1577.
[96] Ghasemi H, Malek-Mohammadi M, Babaie-Zadeh M, et al. Matrix completion based on smoothed rank function[C]//IEEE International Conference on Acoustics, 2011, 3672-3675.
[97] Lai M, Xu Y, Yin W. Improved iteratively reweighted least squares for unconstrained smoothed lq minimization[J]. SIAM Journal on Numerical Analysis, 2013, 51(2):927-957.
[98] Malek-Mohammadi M, Babaie-Zadeh M, Amini A, et al. Recovery of low-rank matrices under affine constraints via a smoothed rank function[J]. IEEE Transactions on Signal Processing, 2014, 62(4):981-992.
[99] Miao W, Pan S, Sun D. A rank-corrected procedure for matrix completion with fixed basis coefficients[J]. Mathematical Programming, 2016, 159(1-2):289-338.
[100] Bi S, Pan S. Multistage convex relaxation approach to rank regularized minimization problems based on equivalent mathematical program with a generalized complementarity constraint[J]. SIAM Journal on Control and Optimization, 2017, 55(4):2493-2518.
[101] Bi S, Pan S, Sun D. A multi-stage convex relaxation approach to noisy structured low-rank matrix recovery[J]. Mathematical Programming Computation, 2020, DOI:10.1007/s12532-020-00177-4.
[102] Le H Y. Generalized subdifferentials of the rank function[J]. Optimization Letters, 2013, 7(4):731-743.
[103] Hiriart-Urruty J B, Le H Y. From Eckart and Young approximation to Moreau envelopes and vice versa[J]. RAIRO-Operations Research, 2013, 47(3):299-310.
[104] Hiriart-Urruty J B, Le H Y. The viscosity subdifferential of the rank function via the corresponding subdifferential of its Moreau envelopes[J]. Acta Mathematica Vietnamica, 2015, 40(4):735-746.
[105] Hiriart-Urruty J B. When only global optimization matters[J]. Journal of Global Optimization, 2013, 56(3):761-763.
[106] Lu Z, Zhang Y, Li X. Penalty decomposition methods for rank minimization[J]. Optimization Methods and Software, 2015, 30(3):531-558.
[107] Vandereycken B, Vandewalle S. A Riemannian optimization approach for computing low rank solutions of Lyapunov equations[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31(5):2553-2579.
[108] Shalit U, Weinshall D, Chechik G. Online learning in the embedded manifold of low-rank matrices[J]. Journal of Machine Learning Research, 2012, 13(Feb):429-458.
[109] Vandereycken B. Low-rank matrix completion by Riemannian optimization[J]. SIAM Journal on Optimization, 2013, 23(2):1214-1236.
[110] Meyer G. Geometric Optimization Algorithms for Linear Regression on Fixed-Rank Matrices[D]. University of Liege, 2011.
[111] Vandereycken B. Riemannian and Multilevel Optimization for Rank-Constrained Matrix Problems with Applications to Lyapunov Equations[D]. Katholieke Universiteit Leuven, 2010.
[112] Ye J. Generalized low rank approximations of matrices[J]. Machine Learning, 2005, 61(1-3):167-191.
[113] Mishra B, Meyer G, Bach F, et al. Low-rank optimization with trace norm penalty[J]. SIAM Journal on Optimization, 2013, 23(4):2124-2149.
[114] Journee M, Bach F, Absil P A, et al. Low-rank optimization on the cone of positive semidefinite matrices[J]. SIAM Journal on Optimization, 2010, 20(5):2327-2351.
[115] Meyer G, Bonnabel S, Sepulchre R. Regression on fixed-rank positive semidefinite matrices:A Riemannian approach[J]. Journal of Machine Learning Research, 2011, 12(Feb):593-625.
[116] Boumal N. Optimization and estimation on manifolds[D]. Louvain:Universite catholique de Louvain, 2014.
[117] Huang W. Optimization algorithms on Riemannian manifolds with applications[D]. Florida:Florida State University, 2013.
[118] Hu J, Liu X, Wen Z, et al. A brief introduction to Manifold optimization[J/OL].[2020-02-01]. Journal of the Operations Research Society of China, https://doi.org/10.1007/s40305-020-00295-9,2020.
[119] Absil P A, Mahony R, Sepulchre R. Optimization Algorithms on Matrix Manifolds[M]. Princeton:Princeton University Press, 2008.
[120] Wen Z, Yin W. A feasible method for optimization with orthogonality constraints[J]. Mathematical Programming, 2013, 142(1-2):397-434.
[121] Gao B, Liu X, Chen X, et al. A new first-order algorithmic framework for optimization problems with orthogonality constraints[J]. SIAM Journal on Optimization, 2018, 28(1):302-332.
[122] Jiang B, Dai Y H. A framework of constraint preserving update schemes for optimization on Stiefel Manifold[J]. Mathematical Programming, 2015, 153(2):535-575.
[123] Gao B, Liu X, Yuan Y. Parallelizable algorithms for optimization problems with orthogonality constraints[J]. SIAM Journal on Scientific Computing, 2019, 41(3):A1949-A1983.
[124] Hu J, Jiang B, Lin L, Wen Z, Yuan Y. Structured quasi-Newton methods for optimization with orthogonality constraints[J]. SIAM Journal on Scientific Computing, 2019, 41(4):A2239-A2269.
[125] Cai T T, Zhou W. Matrix completion via max-norm constrained optimization[J]. Electronic Journal of Statistics, 2016, 10:1493-1525.
[126] Grussler C, Giselsson P. Local convergence of proximal splitting methods for rank constrained problems[C]//IEEE 2017 IEEE 56th Annual Conference on Decision and Control, 2017, 702-708.
[127] Burer S, Monteiro R D. A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization[J]. Mathematical Programming, 2003, 95(2):329-357.
[128] Wen Z, Yin W, Zhang Y. Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm[J]. Mathematical Programming Computation, 2012, 4(4):333-361.
[129] Mardani M, Mateos G, Giannakis G B. Rank minimization for subspace tracking from incomplete data[C]//2013 IEEE International Conference on Acoustics, Speech and Signal Processing, 2013, 5681-5685.
[130] Sun R, Luo Z Q. Guaranteed matrix completion via non-convex factorization[J]. IEEE Transactions on Information Theory, 2016, 62(11):6535-6579.
[131] Zhu Z, Li Q, Tang G, et al. Global optimality in low-rank matrix optimization[C]//2017 IEEE Global Conference on Signal and Information Processing, 2017, 66, 3614-3628.
[132] Li Q, Zhu Z, Tang G. The non-convex geometry of low-rank matrix optimization[J]. Information and Inference:A Journal of the IMA, 2019, 8(1):51-96.
[133] Chi Y, Lu Y M, Chen Y. Nonconvex optimization meets low-rank matrix factorization:An overview[J]. IEEE Transactions on Signal Processing, 2019, 67(20):5239-5269.
[134] Tao P D. Duality in D.C. (difference of convex functions) optimization. subgradient methods[J]. International Series of Numerical Mathematics, 1998, 84(1):277-293.
[135] Li Q, Qi H D. A sequential semismooth Newton method for the nearest low-rank correlation matrix problem[J]. SIAM Journal on Optimization, 2011, 21(4):1641-1666.
[136] Thi H A L, Pham Dinh T, Le H M, et al. DC approximation approaches for sparse optimization[J]. European Journal of Operational Research, 2015, 244(1):26-46.
[137] Bi S, Pan S. Error bounds for rank constrained optimization problems and applications[J]. Operations Research Letters, 2016, 44(3):336-341.
[138] Gotoh J Y, Takeda A, Tono K. DC formulations and algorithms for sparse optimization problems[J]. Mathematical Programming, 2018, 169(1):141-176.
[139] Liu T, Lu Z, Chen X, et al. An exact penalty method for semidefinite-box-constrained low-rank matrix optimization problems[J]. IMA Journal of Numerical Analysis, 2020, 40(1):563-586.
[140] Overton M L, Womersley R S. Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices[J]. Mathematical Programming, 1993, 62(1-3):321-357.
[141] Dattorro J. Convex Optimization and Euclidean Distance Geometry[M]. New York:Meboo Publishing, 2005.
[142] Hempel A B, Goulart P J. A novel method for modelling cardinality and rank constraints[C]//2014 IEEE 53rd Annual Conference on Decision and Control, 2014, 4322-4327.
[143] Delgado R A, Agüero J C, Goodwin G C. A novel representation of rank constraints for real matrices[J]. Linear Algebra and its Applications, 2016, 496:452-462.
[144] Zhou S, Xiu N, Qi H. A fast matrix majorization-projection method for penalized stress minimization with box constraints[J]. IEEE Transactions on Signal Processing, 2018, 66(16):4331-4346.
[145] Bouligand G. Sur les surfaces depourvues de points hyperlimites[J]. Ann. Soc. Polon. Math, 1930, 9:32-41.
[146] Clarke F H. Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations[D]. Washington:University of Washington, 1973.
[147] Mordukhovich B S. Maximum principle in a problem of time optimal control with nonsmooth constraints[J]. Prikladnaia Matematika I Mekhanika, 1976, 40(6):1014-1023.
[148] Rockafellar R T, Wets R J B. Variational Analysis[M]. New York:Springer, 2013.
[149] Attouch H, Jerome B, Benar F S. Convergence of descent methods for semi-algebraic and tame problems:Proximal algorithms, forward-backward splitting, and regularized Gauss Seidel methods[J]. Mathematical Programming, 2013, 137(1-2):91-129.
[150] Pan L, Xiu N, Zhou S. On solutions of sparsity constrained optimization[J]. Journal of the Operations Research Society of China, 2015, 3(4):421-439.
[151] Beck A, Eldar Y C. Sparsity constrained nonlinear optimization:Optimality conditions and algorithms[J]. SIAM Journal on Optimization, 2013, 23(3):1480-1509.
[152] Li X, Xiu N, Zhou S. Matrix optimization over low-rank spectral sets:Stationary points and local and global minimizers[J]. Journal of Optimization Theory and Applications, 2010, 184(3):895-930.
[153] Horn R A, Johnson C R. Matrix Analysis[M]. Cambridge:Cambridge University Press, 2013.
[154] Helmke U, Shayman M A. Critical points of matrix least squares distance functions[J]. Linear Algebra and its Applications, 1995, 215(2):1-19.
[155] Schneider R, Uschmajew A. Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality[J]. SIAM Journal on Optimization, 2015, 25(1):622-646.
