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Limited memory BFGS algorithm for the matrix approximation problem in Frobenius norm. (English) Zbl 1449.65123
Summary: This paper proposes an L-BFGS algorithm based on the active set technique to solve the matrix approximation problem: given a symmetric matrix, find a nearest approximation matrix in the sense of Frobenius norm to make it satisfy some linear equalities, inequalities and a positive semidefinite constraint. The problem is a convex optimization problem whose dual problem is a nonlinear convex optimization problem with non-negative constraints. Under the Slater constraint qualification, it has zero duality gap with the dual problem. To handle large-scale dual problem, we make use of the active set technique to estimate the active constraints, and then the L-BFGS method is used to accelerate free variables. The global convergence of the proposed algorithm is established under certain conditions. Finally, we conduct some preliminary numerical experiments, and compare the L-BFGS method with the inexact smoothing Newton method, the projected BFGS method, the alternating direction method and the two-metric projection method based on the L-BFGS. The numerical results show that our algorithm has some advantages in terms of CPU time when a large number of linear constraints are involved.

##### MSC:
 65K05 Numerical mathematical programming methods 90C55 Methods of successive quadratic programming type 90C30 Nonlinear programming
##### Software:
FEAST; LBFGS-B; TRON; KELLEY; L-BFGS
Full Text:
##### References:
 [1] Borsdorf, R.; Higham, Nj, A preconditioned Newton algorithm for the nearest correlation matrix, IMA J Numer Anal, 30, 94-107 (2010) · Zbl 1188.65055 [2] Boyd, S.; Xiao, L., Least squares covariance matrix adjustment, SIAM J Matrix Anal Appl, 27, 532-546 (2005) · Zbl 1099.65039 [3] Byrd, Rh; Lu, P.; Nocedal, J.; Zhu, C., A limited memory algorithm for bound constrained optimization, SIAM J Sci Comput, 16, 1190-1208 (1995) · Zbl 0836.65080 [4] Cristofari, A.; De Santis, M.; Lucidi, S.; Rinaldi, F., A two-stage active-set algorithm for bound-constrained optimization, J Optim Theory Appl, 172, 2, 369-401 (2017) · Zbl 1398.90170 [5] Dykstra, Rl, An algorithm for restricted least squares regression, J Am Stat Assoc, 78, 837-842 (1983) · Zbl 0535.62063 [6] FEAST solver (2009-2015). http://www.feast-solver.org/. Accessed 17 June 2018 [7] Gabay, D.; Fortin, M.; Glowinski, R., Application of the method of multipliers to variational inequalities, Augmented Lagrangian methods: application to the numerical solution of boundary-value problems, 299-331 (1983), Amsterdam: North-Holland, Amsterdam [8] Gabay, D.; Mercier, B., A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput Math Appl, 2, 17-40 (1976) · Zbl 0352.65034 [9] Gafni, Em; Bertsekas, Dp, Two-metric projection methods for constrained optimization, SIAM J Control Optim, 22, 6, 936-964 (1984) · Zbl 0555.90086 [10] Gao, Y.; Sun, Df, Calibrating least squares semidefinite programming with equality and inequality constraints, SIAM J Matrix Anal Appl, 31, 1432-1457 (2009) · Zbl 1201.49031 [11] Hager, Ww; Zhang, H., A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J Optim, 16, 170-192 (2005) · Zbl 1093.90085 [12] Hager, Ww; Zhang, H., A new active set algorithm for box constrained optimization, SIAM J Optim, 17, 526-557 (2006) · Zbl 1165.90570 [13] Han, Rq; Xie, Wj; Xiong, X.; Zhang, W.; Zhou, Wx, Market correlation structure changes around the great crash: a random matrix theory analysis of the Chinese stock market, Fluct Noise Lett, 16, 2, 1750018 (2017) [14] He, Bs; Xu, Mh; Yuan, Xm, Solving large-scale least squares semidefinite programming by alternating direction methods, SIAM J Matrix Anal Appl, 32, 136-152 (2011) · Zbl 1243.49039 [15] Higham, Nj, Computing the nearest correlation matrix-a problem from finance, IMA J Numer Anal, 22, 329-343 (2002) · Zbl 1006.65036 [16] Kelley, Ct, Iterative methods for optimization, 102-104 (1999), Philadelphia: SIAM, Philadelphia [17] Kupiec, Ph, Stress testing in a value-at-risk framework, J Deriv, 6, 7-24 (1998) [18] Li, Qn; Li, Dh, A projected semismooth Newton method for problems of calibrating least squares covariance matrix, Oper Res Lett, 39, 103-108 (2011) · Zbl 1218.90220 [19] Lin, C-J; Moré, Jj, Newtons method for large bound-constrained optimization problems, SIAM J Optim, 9, 1100C1127 (1999) · Zbl 0957.65064 [20] Liu, Dc; Nocedal, J., On the limited memory BFGS method for large-scale optimization, Math Program, 45, 503-528 (1989) · Zbl 0696.90048 [21] Malick, J., A dual approach to semidefinite least squares problems, SIAM J Matrix Anal Appl, 26, 272-284 (2004) · Zbl 1080.65027 [22] Ni, Q.; Yuan, Y., A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization, Math Comp, 66, 1509-1520 (1997) · Zbl 0886.65065 [23] Nocedal, J.; Wright, Sj, Numerical optimization (2006), New York: Springer, New York [24] Polanco-Martínez, Jm; Fernández-Macho, J.; Neumann, Mb; Faria, Sh, A pre-crisis vs. crisis analysis of peripheral EU stock markets by means of wavelet transform and a nonlinear causality test, Phys A, 490, 1211-1227 (2018) [25] Polizzi, E., Density-matrix-based algorithms for solving eigenvalue problems, Phys Rev B, 79, 115112 (2009) [26] Qi, Hd; Sun, D., A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM J Matrix Anal Appl, 28, 360-385 (2006) · Zbl 1120.65049 [27] Qi, Hd; Sun, D., Correlation stress testing for value-at-risk: an unconstrained convex optimization approach, Comput Optim Appl, 45, 427-462 (2010) · Zbl 1198.91091 [28] Rahpeymaii, F.; Kimiaei, M.; Bagheri, A., A limited memory quasi-Newton trust-region method for box constrained optimization, J Comput Appl Math, 303, 105-118 (2016) · Zbl 1381.90097 [29] Rebonato, R.; Jäckel, P., The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes, J Risk, 2, 17-27 (1999) [30] Rockafellar RT (1974) Conjugate duality and optimization. BMS-NSF Regional Conference Series in Applied Mathematics 16, SIAM, Philadelphia · Zbl 0296.90036 [31] Schwertman, Nc; Allen, Dm, Smoothing an indefinite variance-covariance matrix, J Stat Comput Simul, 9, 183-194 (1979) [32] So, Mkp; Wong, J.; Asai, M., Stress testing correlation matrices for risk management, N Am J Econ Finan, 26, 310-322 (2013) [33] Sun, Yf; Vandenberghe, L., Decomposition methods for sparse matrix nearness problems, SIAM J Matrix Anal Appl, 36, 1691-1717 (2015) · Zbl 1342.90128 [34] Werner R, Schöttle K (2007) Calibration of correlation matrices-SDP or not SDP. Technical report, Munich University of Technology, Munich [35] Xiao, Y.; Wei, Z., A new subspace limited memory BFGS algorithm for large-scale bound constrained optimization, Appl Math Comput, 185, 1, 350-359 (2007) · Zbl 1114.65069 [36] Xiao, Y.; Li, Dh, An active set limited memory BFGS algorithm for large-scale bound constrained optimization, Math Methods Oper Res, 67, 3, 443-454 (2008) · Zbl 1145.90084 [37] Xue WJ, Shen CG (2018) Limited memory BFGS method for least squares semidefinite programming with banded structure. Technical Report, University of Shanghai for Science and Technology, Shanghai [38] Yuan, G.; Lu, X., An active set limited memory BFGS algorithm for bound constrained optimization, Appl Math Model, 35, 7, 3561-3573 (2011) · Zbl 1221.90082 [39] Yuan, G.; Wei, Z., Convergence analysis of a modified BFGS method on convex minimizations, Comput Optim Appl, 47, 237-255 (2010) · Zbl 1228.90077 [40] Ye, Ch; Yuan, Xm, A descent method for structured monotone variational inequalities, Optim Methods Softw, 22, 329-338 (2007) · Zbl 1196.90118 [41] Zarantonello, Eh; Zarantonello, Eh, Projections on convex sets in Hilbert space and spectral theory I and II, Contributions to nonlinear functional analysis, 237-424 (1971), New York: Academic Press, New York [42] Zhang, Hc; Hager, Ww, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J Optim, 14, 1043-1056 (2004) · Zbl 1073.90024
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