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Matrix optimization over low-rank spectral sets: stationary points and local and global minimizers. (English) Zbl 1432.90124
Summary: In this paper, we consider matrix optimization with the variable as a matrix that is constrained into a low-rank spectral set, where the low-rank spectral set is the intersection of a low-rank set and a spectral set. Three typical spectral sets are considered, yielding three low-rank spectral sets. For each low-rank spectral set, we first calculate the projection of a given point onto this set and the formula of its normal cone, based on which the induced stationary points of matrix optimization over low-rank spectral sets are then investigated. Finally, we reveal the relationship between each stationary point and each local/global minimizer.
90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
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[1] Recht, B.; Fazel, M.; Parrilo, Pa, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52, 3, 471-501 (2010) · Zbl 1198.90321
[2] Markovsky, I., Low Rank Approximation (2012), London: Springer, London · Zbl 1245.93005
[3] Ding, C.: An introduction to a class of matrix optimization problems. Ph.D. thesis, National University of Singapore (2012)
[4] Udell, M.; Horn, C.; Zadeh, Rb; Boyd, Sp, Generalized low rank models, Mach. Learn., 9, 1, 1-118 (2016) · Zbl 1350.68221
[5] Davenport, Ma; Romberg, J., An overview of low-rank matrix recovery from incomplete observations, IEEE J. Sel. Top. Signal Process., 10, 4, 608-622 (2016)
[6] Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University (2002)
[7] Candès, Ej; Tao, T., The power of convex relaxation: near-optimal matrix completion, IEEE Trans. Inf. Theory, 56, 5, 2053-2080 (2010) · Zbl 1366.15021
[8] Marjanovic, G.; Solo, V., On \(l_q\) optimization and matrix completion, IEEE Trans. Signal Process., 60, 11, 5714-5724 (2012) · Zbl 1393.94353
[9] Mohan, K.; Fazel, M., Iterative reweighted algorithms for matrix rank minimization, J. Mach. Learn. Res., 13, Nov, 3441-3473 (2012) · Zbl 1436.65055
[10] Nie, F., Wang, H., Cai, X., Huang, H., Ding, C.: Robust matrix completion via joint schatten p-norm and \(l_p\)-norm minimization. In: IEEE International Conference on Data Mining, pp. 566-574 (2012)
[11] Chen, Y.; Xiu, N.; Peng, D., Global solutions of non-lipschitz \(s_2-s_p\) minimization over the positive semidefinite cone, Optim. Lett., 8, 7, 2053-2064 (2014) · Zbl 1326.90063
[12] Zhao, Yb, An approximation theory of matrix rank minimization and its application to quadratic equations, Linear Algebra Appl., 437, 1, 77-93 (2012) · Zbl 1242.65086
[13] Yao, H.; Debing, Z.; Jieping, Y.; Xuelong, L.; Xiaofei, H., Fast and accurate matrix completion via truncated nuclear norm regularization, IEEE Trans. Pattern Anal. Mach. Intell., 35, 9, 2117-2130 (2013)
[14] Burer, S.; Monteiro, Rdc, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program., 95, 2, 329-357 (2003) · Zbl 1030.90077
[15] Journée, M.; Bach, F.; Absil, Pa; Sepulchre, R., Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. Optim., 20, 5, 2327-2351 (2010) · Zbl 1215.65108
[16] Wen, Z.; Yin, W.; Zhang, Y., Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4, 4, 333-361 (2012) · Zbl 1271.65083
[17] Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 665-674. ACM (2013) · Zbl 1293.65073
[18] Hardt, M.: Understanding alternating minimization for matrix completion. In: 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 651-660. IEEE (2014)
[19] Sun, R.; Luo, Z., Guaranteed matrix completion via non-convex factorization, IEEE Trans. Inf. Theory, 62, 11, 6535-6579 (2016) · Zbl 1359.94179
[20] Gao, Y.: Structured low rank matrix optimization problems: A penalty approach. Ph.D. thesis, National University of Singapore (2010)
[21] Kim, Sj; Moon, Yh, Structurally constrained \({H}_2\) and \({H}_{\infty }\) control: a rank-constrained LMI approach, Automatica, 42, 9, 1583-1588 (2006) · Zbl 1128.93325
[22] Delgado, Ra; Agüero, Jc; Goodwin, Gc, A rank-constrained optimization approach: application to factor analysis, IFAC Proc. Vol., 47, 3, 10373-10378 (2014)
[23] Bi, S.; Pan, S., Error bounds for rank constrained optimization problems and applications, Oper. Res. Lett., 44, 3, 336-341 (2016) · Zbl 1408.90279
[24] Zhou, S.; Xiu, N.; Qi, H., A fast matrix majorization-projection method for penalized stress minimization with box constraints, IEEE Trans. Signal Process., 66, 16, 4331-4346 (2018) · Zbl 1414.90260
[25] Luke, Dr, Prox-regularity of rank constraint sets and implications for algorithms, J. Math. Imaging Vis., 47, 3, 231-238 (2013) · Zbl 1311.90142
[26] Rockafellar, Rt; Wets, Rj, Variational Analysis (1998), Berlin: Springer, Berlin
[27] Cason, Tp; Absil, Pa; Dooren, Pv, Iterative methods for low rank approximation of graph similarity matrices, Linear Algebra Appl., 438, 4, 1863-1882 (2013) · Zbl 1264.65060
[28] Schneider, R.; Uschmajew, A., Convergence results for projected line-search methods on varieties of low-rank matrices via łojasiewicz inequality, SIAM J. Optim., 25, 1, 622-646 (2015) · Zbl 1355.65079
[29] Zhou, G.; Huang, W.; Gallivan, Ka; Van Dooren, P.; Absil, Pa, A Riemannian rank-adaptive method for low-rank optimization, Neurocomputing, 192, 72-80 (2016)
[30] Li, X.; Song, W.; Xiu, N., Optimality conditions for rank-constrained matrix optimization, J. Oper. Res. Soc. China, 7, 2, 285-301 (2019) · Zbl 1438.90272
[31] Linial, N.; London, E.; Rabinovich, Y., The geometry of graphs and some of its algorithmic applications, Combinatorica, 15, 2, 215-245 (1995) · Zbl 0827.05021
[32] Biswas, P., Ye, Y.: Semidefinite programming for ad hoc wireless sensor network localization. In: International Symposium on Information Processing in Sensor Networks (2004) · Zbl 1100.90029
[33] Ji, S., Sze, K.F., Zhou, Z., So, M.C., Ye, Y.: Beyond convex relaxation: a polynomial-time non-convex optimization approach to network localization. In: IEEE Infocom (2013)
[34] Borsdorf, R.; Higham, Nj; Raydan, M., Computing a nearest correlation matrix with factor structure, SIAM J. Matrix Anal. Appl., 31, 5, 2603-2622 (2010) · Zbl 1213.65022
[35] Higham, Nj, Computing the nearest correlation matrix a problem from finance, IMA J. Numer. Anal., 22, 3, 329-343 (2018) · Zbl 1006.65036
[36] Dukanovic, I.; Rendl, F., Semidefinite programming relaxations for graph coloring and maximal clique problems, Math. Program., 109, 2-3, 345-365 (2007) · Zbl 1278.90299
[37] Kalev, A.; Kosut, Rl; Deutsch, Ih, Quantum tomography protocols with positivity are compressed sensing protocols, Nat. Partn. J. Quantum Inf., 1, 1, 15018 (2015)
[38] Lewis, As, Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl., 17, 4, 927-949 (1996) · Zbl 0876.15016
[39] Tam, Mk, Regularity properties of non-negative sparsity sets, J. Math. Anal. Appl., 447, 2, 758-777 (2017) · Zbl 1353.15030
[40] Lu, Z.; Zhang, Y.; Li, X., Penalty decomposition methods for rank minimization, Optim. Methods Softw., 30, 3, 531-558 (2015) · Zbl 1323.65070
[41] Kyrillidis, A.: Rigorous optimization recipes for sparse and low rank inverse problems with applications in data sciences. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2014)
[42] Mordukhovich, Bs, Variational Analysis and Generalized Differentiation I: Basic Theory (2006), Berlin: Springer, Berlin
[43] Hiriart-Urruty, Jb; Lemaréchal, C., Fundamentals of Convex Analysis (2012), Berlin: Springer, Berlin
[44] Drusvyatskiy, D.; Kempton, C., Variational analysis of spectral functions simplified, J. Convex Anal., 25, 1, 119-134 (2018) · Zbl 1398.49012
[45] Lu, Z.: Optimization over sparse symmetric sets via a nonmonotone projected gradient method (2015). arXiv preprint arXiv:1509.08581
[46] Pan, L.; Xiu, N.; Fan, J., Optimality conditions for sparse nonlinear programming, Sci. China Math., 60, 5, 1-18 (2017) · Zbl 1365.90220
[47] Beck, A.; Eldar, Yc, Sparsity constrained nonlinear optimization: optimality conditions and algorithms, SIAM J. Optim., 23, 3, 1480-1509 (2013) · Zbl 1295.90051
[48] Horn, Ra; Johnson, Cr, Matrix Analysis (2013), New York: Cambridge University Press, New York
[49] Pan, L.; Zhou, S.; Xiu, N.; Qi, Hd, A convergent iterative hard thresholding for nonnegative sparsity optimization, Pac. J. Optim., 13, 2, 325-353 (2017) · Zbl 1384.90078
[50] Bertsekas, Dp, Nonlinear Programming (1999), Belmont: Athena Scientific, Belmont
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