Abstract
We develop in this paper a method based on a d.c. (difference of convex functions) optimization approach called DCA for solving large-scale exact distance geometry problem. Requiring only matrix-vector products and one Cholesky factorization, the DCA seems to be robust and efficient in large scale problems. Moreover it allows exploiting sparsity of the given distance matrix. A technique using the triangle inequality to generate a complete approximate distance matrix was investigated in order to compute a good starting point for the DCA. Numerical simulations of the molecular conformation problems with up to 12288 variables are reported which prove the robustness, the efficiency, and the globality of our algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blumenthal, L.M. (1953) Theory and Applications of Distance Geometry. Oxford University Press
Crippen, G.M., Havel, T.F. (1998) Distance Geometry and Molecular Conformation. John Wiley&Sons
De Leeuw, J. (1977) Applications of convex analysis to multidimensional scaling. In: Barra J.R. et al. (Eds.) Recent developments in statistics, North-Holland Publishing Company, 133–145
Glun, W., Hayden, T.L., Raydan, M.(1993) Molecular Conformation from distance matrices. J. Comp. Chem. 14, 114–120
Havel, T.F. (1991) An evaluation of computational strategies for use in the determination of protein structure from distance geometry constraints obtained by nuclear magnetic resonance. Prog. Biophys. Mol. Biol. 56, 43–78
Hendrickson, B.A. (1995) The molecule problem: Exploiting structure in global optimization. SIAM J. Optimization 5, 835–857.
Le Thi Hoai An (1997) Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algorithmes et Applications. Habilitation à Diriger des Recherches, Université de Rouen
Le Thi Hoai An, Pham Dinh Tao (1997) Solving a class of linearly constrained indefinite quadratic problems by D.c. algorithms. Journal of Global Optimization 11, 253–285
Moré, J.J., Wu, Z. (1997) Global continuation for distance geometry problems. SIAM J. Optimization 8, 814–836
Moré, J.J., Wu, Z. (1996) Issues in large-scale Molecular Optimization. Preprint MCS-P539-1095, Argonne National Laboratory, Argonne, Illinois 60439
Pham Dinh Tao, Souad El. B. (1986) Algorithms for solving a class of non convex optimization problems. Methods of subgradients. In: Hiriart Urruty J.B. (Ed.) Fermat days 85. Mathematics for Optimization. Elsevier Science Publishers B.V., North-Holland, 249–271
Pham Dinh Tao, Souad El. B. (1988) Duality in d.c. (difference of convex functions) optimization. Subgradient methods. Trends in Mathematical Optimization, International Series of Numer Math. 84, Birkhauser, 277–293
Pham Dinh Tao, Le Thi Hoai An (1994) Stabilité de la dualité lagrangienne en optimisation d.c. (différence de deux fonctions convexes) C.R. Acad. Paris 318, Série I, 79–384
Pham Dinh Tao, Le Thi Hoai An (1998) D.c. optimization algorithms for trust region problem. SIAM J. Optimization 8, No 2, 476–505
Pham Dinh Tao, Le Thi Hoai An (1997) Convex analysis approach to d.c. programming: Theory, Algorithms and Applications (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday). Acta Mathematica Vietnamica 22, No 1, 289–355
Rockafellar, R.T. (1970) Convex Analysis. Princeton University, Princeton
Saxe, J. B. (1979) Embeddability of Weighted Graphs in k-space is Strongly NP-hard. Proc. 17 Allerton Conference in Communications, Control and Computing, 480–489
Varga, R. (1962) Matrix iterative analysis. Prentice Hall
Zou, Z., Richard H. Bird, Schnabel, R.B. (1997) A Stochastic/Pertubation Global Optimization Algorithm for Distance Geometry Problems. J. of Global Optimization 11, 91–105
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
An, L.T.H., Tao, P.D. (2000). Large Scale Molecular Conformation via the Exact Distance Geometry Problem. In: Nguyen, V.H., Strodiot, JJ., Tossings, P. (eds) Optimization. Lecture Notes in Economics and Mathematical Systems, vol 481. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57014-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-57014-8_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66905-0
Online ISBN: 978-3-642-57014-8
eBook Packages: Springer Book Archive