Abstract
Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-known assignment problem.
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Balogh, J., Csendes, T. and Rapcsak, T. (2002), Global optimization on Stiefel manifolds, NMCM-2002 Book of Abstracts, Miskolc, Hungary, 19–21.
Bolla, M., Michaletzky, G., Tusnady, G. and Ziermann, M. (1998), Extrema of sums of heterogeneous quadratic forms, Linear Algebra and its Applications 269, 331–365.
Casado, L.G., Garcìa, I. and Csendes, T. (2002), A new multisection technique in interval methods for global optimization. Computing 65, 263–269.
Corliss, G.F. and Kearfott, R.B. (1999), Rigorous global search: Industrial applications. In: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer, Dordrecht, pp. 1–16.
Csendes, T. (1998), Nonlinear parameter estimation by global optimization—efficiency and reliability, Acta Cybernetica 8, 361–370.
Floudas, C.A. and Pardalos, P.M. (1990), A Collection of Test Problems for Constrained Global Optimization Algorithms, Lecture Notes in Computer Science 455, Springer-Verlag, Berlin.
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A. and Schweiger, C.A. (1999), Handbook of Test Problems for Local and Global Optimization, Kluwer, Dordrecht. 277–290.
Horst, R. and Pardalos, P.M. (1995), Handbook of Global Optimization, Kluwer, Dordrecht.
Kearfott, R.B. (1996), Rigorous Global Search: Continuous Problems, Kluwer, Dordrecht.
MarkÓt, M. Cs., J. Fernandez, L.G. Casado´ and T. Csendes (submitted for publication), New interval methods for constrained optimization. Available at http://www.inf.u-szeged.hu/ ~markot/.
Pardalos, P., Rendl, F. and Wolkowicz, H. (1994), The quadratic assignment problem: A survey and recent developments, Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, AMS, pp. 1–41.
Rapcsak, T. (2001), On minimization of sums of heterogeneous quadratic functions on Stiefel manifolds, In. Pardalos, P., Migdalas, A. and Varbrand, P. (eds.), From local to global optimization, Kluwer, Dordrecht, pp. 277–290.
Rapcsak, T. (2002), On minimization on Stiefel manifolds, European Journal of Operational Research 143, 365–376.
Rapcsak, T. (2004), Some optimization problems in statistics, Journal of Global Optimization 28, 217–228.
Sahni, S. and Gonzalez, T. (1996), P-complete approximation problems, Journal of the Association for Computing Machinery 23, 555–565.
Stiefel, E. (1935–1936), Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Commentarii Math. Helvetici 8, 305–353.
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Balogh, J., Csendes, T. & Rapcsák, T. Some Global Optimization Problems on Stiefel Manifolds. Journal of Global Optimization 30, 91–101 (2004). https://doi.org/10.1007/s10898-004-0574-9
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DOI: https://doi.org/10.1007/s10898-004-0574-9