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Non-monotone algorithm for minimization on arbitrary domains with applications to large-scale orthogonal Procrustes problem. (English) Zbl 1354.65120
Summary: This paper concerns a non-monotone algorithm for minimizing differentiable functions on closed sets. A general numerical scheme is proposed which combines a regularization/trust-region framework with a non-monotone strategy. Global convergence to stationary points is proved under usual assumptions. Numerical experiments for a particular version of the general algorithm are reported. In addition, a promising numerical scheme for medium/large-scale orthogonal Procrustes problem is also proposed and numerically illustrated.
Reviewer: Reviewer (Berlin)

MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C51 Interior-point methods
Software:
CUTEst; GALAHAD; na26
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[1] Arias, T. A.; Edelman, A.; Smith, S. T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 303-353, (1998) · Zbl 0928.65050
[2] Baglama, J.; Reichel, L., Restarted block Lanczos bidiagonalization methods, Numer. Algorithms, 43, 251-272, (2006) · Zbl 1110.65027
[3] Baglama, J.; Reichel, L., An implicitly restarted block Lanczos bidiagonalization method using Leja shifts, BIT Numer. Math., 53, 285-310, (2013) · Zbl 1269.65038
[4] Barzilai, J.; Borwein, J. M., Two point step size gradient methods, IMA J. Numer. Anal., 8, 141-148, (1988) · Zbl 0638.65055
[5] Bell, T., Global positioning system-based attitude determination and the orthogonal procrustes problem, J. Guid. Control Dyn., 26, 820-822, (2003)
[6] Birgin, E. G.; Martínez, J. M.; Raydan, M., Inexact spectral projected gradient methods on convex sets, IMA J. Numer. Anal., 23, 539-559, (2003) · Zbl 1047.65042
[7] Bojanczyk, A. W.; Lutoborski, A., The procrustes problem for orthogonal Stiefel matrices, SIAM J. Sci. Comput., 21, 1291-1304, (1999) · Zbl 0962.65037
[8] Chu, M. T.; Trendafilov, N. T., The orthogonally constrained regression revisited, J. Comput. Graph. Stat., 10, 746-771, (2001)
[9] Cui, Z.; Wu, B., A new modified nonmonotone adaptive trust region method for unconstrained optimization, Comput. Optim. Appl., 53, 795-806, (2012) · Zbl 1264.90158
[10] Dai, Y. H., On the nonmonotone line search, J. Optim. Theory Appl., 112, 315-330, (2002) · Zbl 1049.90087
[11] Dolan, E. D.; Moré, J. J., Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213, (2002) · Zbl 1049.90004
[12] Francisco, J. B.; Viloche Bazán, F. S., Nonmonotone algorithm for minimization on closed sets with application to minimization on Stiefel manifolds, J. Comput. Appl. Math., 236, 2717-2727, (2012) · Zbl 1239.65035
[13] Fu, J.; Sun, W., Nonmonotone adaptive trust-region method for unconstrained optimization problems, Appl. Math. Comput., 163, 489-504, (2005) · Zbl 1069.65063
[14] Golub, G. H.; Van Loan, C. F., Matrix computations, (1996), The Johns Hopkins University Press London · Zbl 0865.65009
[15] Gould, N. I.M.; Orban, D.; Toint, P. L., Galahad, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization, ACM Trans. Math. Softw., 29, 353-372, (2004) · Zbl 1068.90525
[16] Gould, N. I.M.; Orban, D.; Toint, P. L., Cutest: a constrained and unconstrained testing environment with safe threads, (2013), Rutherford Appleton Laboratory, Technical Report RAL-TR-2013-005 · Zbl 1325.90004
[17] Gower, J., Orthogonal and projection procrustes analysis, (Krzanowski, W., Recent Advances in Descriptive Multivariate Statistics, (1995), Oxford University Press Oxford), 113-134
[18] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal., 23, 707-716, (1986) · Zbl 0616.65067
[19] Grippo, L.; Lampariello, F.; Lucidi, S., A class of nonmonotone stabilization methods in unconstrained optimization, Numer. Math., 59, 779-805, (1991) · Zbl 0724.90060
[20] Higham, N. J., The symmetric procrustes problem, BIT Numer. Math., 28, 133-143, (1988) · Zbl 0641.65034
[21] Karimi, S.; Toutounian, F., The block least squares method for solving nonsymmetric linear systems with multiples right-hand sides, Appl. Math. Comput., 177, 852-862, (2006) · Zbl 1096.65040
[22] Mo, J.; Liu, C.; Yan, S., A nonmonotone trust region method based on nonincreasing technique of weighted average of successive function values, J. Comput. Appl. Math., 209, 97-108, (2007) · Zbl 1142.65049
[23] Nemirovski, A., Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Math. Program., 109, 283-317, (2007) · Zbl 1156.90012
[24] Ortega, J. M.; Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[25] Rapcsák, T., On minimization on Stiefel manifolds, Eur. J. Oper. Res., 143, 365-376, (2002) · Zbl 1058.90064
[26] Raydan, M., The Barzilai and Borwein gradient method for large scale unconstrained minimization problem, SIAM J. Optim., 7, 26-33, (1997) · Zbl 0898.90119
[27] Shapiro, A.; Al-Khayyal, F., First-order conditions for isolated locally optimal solutions, J. Optim. Theory Appl., 77, 189-196, (1993) · Zbl 0792.90074
[28] Sun, W., Nonmonotone trust region method for solving optimization problems, Appl. Math. Comput., 156, 159-174, (2004) · Zbl 1059.65055
[29] Sun, W. Y.; Han, J. Y.; Sun, J., Global convergence of non-monotone descent methods for unconstrained optimization problems, J. Comput. Appl. Math., 146, 89-98, (2002) · Zbl 1007.65044
[30] Toint, P. L., An assessment of non-monotone linesearch techniques for unconstrained optimization, SIAM J. Sci. Comput., 17, 725-739, (1996) · Zbl 0849.90113
[31] Toint, P. L., Non-monotone trust region algorithm for nonlinear optimization subject to convex constraints, Math. Program., 77, 69-94, (1997) · Zbl 0891.90153
[32] Trendedafilov, N. T., On the \(l_1\) procrustes problem, Future Gener. Comput. Syst., 19, 1177-1186, (2003)
[33] Wen, Z.; Yin, W., A feasible method for optimization with orthogonality constraints, (2010), Rice University CAAM, Rice University, Technical Report TR10-26
[34] Yu, Z., Solving bound constrained optimization via a new nonmonotone spectral projected gradient method, Appl. Numer. Math., 58, 1340-1348, (2008) · Zbl 1154.65051
[35] Zhang, H.; Hager, W., A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14, 1043-1056, (2004) · Zbl 1073.90024
[36] Zhang, Z.; Du, K., Successive projection method for solving the unbalanced procrustes problem, Sci. China Ser. A, 49, 971-986, (2006) · Zbl 1112.65039
[37] Zhou, Q.; Hang, D., Nonmonotone adaptive trust region method with line search based on new diagonal updating, Appl. Numer. Math., 91, 75-88, (2015) · Zbl 1310.65070
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