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D.C. Programming Approach for Large-Scale Molecular Optimization via the General Distance Geometry Problem

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Book cover Optimization in Computational Chemistry and Molecular Biology

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 40))

Abstract

In this paper we are concerned with a new d.c.(difference of convex functions) approach to the general distance geometry problem and the two phase solution algorithm DCA. We present a thorough study of this d.c. program in its elegant matrix formulation including substantial subdifferential calculus for related convex functions. It makes it possible to express DCA in its simplest form and to exploit sparsity. In Phase 1 we extrapolate all pairwise dissimilarities from given bound constraints and then apply DCA to the resulting Euclidean Multidimensional Scaling (EMDS) problem. In Phase 2 we solve the original problem by applying DCA from the point obtained by Phase 1. Requiring only matrix-vector products and one Cholesky factorization, DCA seems to be robust and efficient in the large scale setting as proved by numerical simulations which furthermore indicated that DCA always converges to global solutions.

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References

  1. Abdo Y. Alfakih, A. Khandani & H. Wolkowicz, An interior-point method for the Euclidean distance matrix completion problem, Research Report CORR 97–9, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.

    Google Scholar 

  2. Le Thi Hoai An, Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des Recherches, Université de Rouen, Juin 1997.

    Google Scholar 

  3. Le Thi Hoai An and Pham Dinh Tao (1997), Solving a class of linearly constrained indefinite quadratic problems by D.c. algorithms, Journal of Global Optimization, 11 pp. 253–285.

    Article  MATH  Google Scholar 

  4. Le Thi Hoai An and Pham Dinh Tao (1998) A Branch-and-Bound method via D. C. Optimization Algorithm and Ellipsoidal technique for Box Constrained Nonconvex Quadratic Programming Problems, Journal of Global Optimization 13(1998), pp. 171–206.

    Article  MATH  Google Scholar 

  5. Le Thi Hoai An and Pham Dinh Tao (1999) A continuous approach for large-scale linearly constrained quadratic zero-one programming, To appear in Optimization.

    Google Scholar 

  6. Le Thi Hoai An, Pham Dinh Tao and L.D. Muu (1996), Numerical solution for Optimization over the efficient set by D.c. Optimization Algorithm, Operations Research Letters, 19, pp. 117–128.

    Article  MathSciNet  MATH  Google Scholar 

  7. L.M. Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press, 1953.

    MATH  Google Scholar 

  8. G.M. Crippen, Rapid calculation of coordinates from distance measures, J. Computational Physics, Vol 26 (1978), pp. 449–452.

    Article  MATH  Google Scholar 

  9. G.M. Crippen & T.F. Havel, Distance Geometry and Molecular Conformation, John Wiley &; Sons, 1988

    MATH  Google Scholar 

  10. J. De Leeuw, Applications of convex analysis to multidimensional scaling, Recent developments in Statistics, J.R. Barra et al., editors, North-Holland Publishing Company, pp. 133–145, 1977.

    Google Scholar 

  11. J. De Leeuw, Convergence of the majorization method for multidimansional scaling, Journal of Classification, Vol 5 (1988), pp. 163–180.

    Article  MathSciNet  MATH  Google Scholar 

  12. V.F. Demyanov & L.V. Vasilev, Nondifferentiable optimization, Optimization Software, Inc. Publications Division, New York, (1985)

    Book  Google Scholar 

  13. T.F. Havel, 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 (1991), pp. 43–78.

    Article  Google Scholar 

  14. W. Glunt, T.L. Hayden & M. Raydan, Molecular Conformation from distance matrices, J. Comp. Chem., 14 (1993), pp. 114–120.

    Article  Google Scholar 

  15. L. Guttman, A general nonmetric technique for finding the smallest coordinate space for a configuration of point, Psychometrika, Vol 33 (1968), pp. 469–506.

    Article  MATH  Google Scholar 

  16. B.A. Hendrickson, The molecule problem: Exploiting structure in global optimization, SIAM J. Optimization, 5(1995), pp. 835–857.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.B. Hiriart Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer-Verlag Berlin Heidelberg, 1993.

    Google Scholar 

  18. P.J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972.

    MATH  Google Scholar 

  19. P. Mahey, Nguyen Van Hien & Pham Dinh Tao, Proximal techniques for convex programming, Technical Report, Lab ARTEMIS, IMAG, CNRS-Grenoble, 1992.

    Google Scholar 

  20. P. Mahey and Pham Dinh Tao, Partial regularization of the sum of two maximal monotone operators, Math. Modell. Numer. Anal. M 2 AN, Vol. 27 (1993), pp. 375–395.

    MATH  Google Scholar 

  21. P. Mahey and Pham Dinh Tao, Proximal decomposition of the graph of maximal monotone operator, SIAM J. Optim. 5 (1995), pp. 454–468.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Laurent, Cuts, matrix completions and a graph rigidity, Mathematical Programming, Vol. 79 (1997), Nos 1–3, pp. 255–283.

    Google Scholar 

  23. J.J. More & Z. Wu, Global continuation for distance geometry problems, SIAM J. Optimization, 8(1997), pp. 814–836.

    Article  MathSciNet  Google Scholar 

  24. J.J. More & Z. Wu, Issues in large-scale Molecular Optimization, preprint MCS-P539–1095, Argonne National Laboratory, Argonne, Illinois 60439, Mars 1996.

    Google Scholar 

  25. J.J. More & Z. Wu, Distance geometry optimization for protein structures, preprint MCS-P628–1296, Argonne National Laboratory, Argonne, Illinois 60439, December 1996.

    Google Scholar 

  26. A.M. Ostrowski, Solutions of equations and systems of equations, Academic Press, New York, 1966.

    Google Scholar 

  27. Pham Dinh Tao, Eléments homoduaux d’une matrice A relatifs à un couple de normes (ø,Ψ). Applications au calcul de SøΨ(A), Séminaire d’Analyse Numérique, Grenoble, n° 236,1975

    Google Scholar 

  28. Pham Dinh Tao, Calcul du maximum d’une forme quadratique définie positive sur la boule unité de la norme du maximum, Séminaire d’Analyse Numérique, Grenoble, n°247,1976

    Google Scholar 

  29. Pham Dinh Tao, Contribution à la théorie de normes et ses applications àl’analyse numérique, Thèse de Doctorat d’Etat Es Science, Université Joseph Fourier- Grenoble, 1981.

    Google Scholar 

  30. Pham Dinh Tao, Convergence of subgradient method for computing the bound-norm of matrices, Linear Alg. and Its Appl., Vol 62 (1984), pp. 163–182.

    Article  MATH  Google Scholar 

  31. Pham Dinh Tao, Algorithmes de calcul d’une forme quadratique sur la boule unité de la norme maximum, Numer. Math., Vol 45 (1985), pp. 377–440.

    Article  Google Scholar 

  32. Pham Dinh Tao, Algorithms for solving a class of non convex optimizationproblems. Methods of subgradients, Fermat days 85. Mathematics for Optimization, Elsevier Science Publishers B.V. NorthHolland, (1986), pp. 249–271.

    Google Scholar 

  33. Pham Dinh Tao, Duality in d.c. (difference of convex functions) optimization.Subgradient methods, Trends in Mathematical Optimization, International Series of Numer Math. Vol 84 (1988), Birkhauser, pp. 277–293.

    Google Scholar 

  34. Pham Dinh Tao and Le Thi Hoai An, D.c. optimization algorithms for the trust region problem. SIAM J. Optimization, Vol. 8, No 2, (1998), pp. 476–505.

    Google Scholar 

  35. Pham Dinh Tao and Le Thi Hoai An, 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, Vol. 22, No 1, 1997, pp 289–355.

    MathSciNet  MATH  Google Scholar 

  36. Pham Dinh Tao and Le Thi Hoai An, D.C. Programming approach and solution algorithm to the Multidimensional Scaling Problem, To appear.

    Google Scholar 

  37. R.T. Rockafellar, Convex Analysis, Princeton University, Princeton, 1970.

    MATH  Google Scholar 

  38. R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control and Optimization, Vol.14, N°5 (1976)

    Google Scholar 

  39. J. B. Saxe, Embeddability of Weighted Graphs in k-space is Strongly NP-hard, Proc. 17 Allerton Conference in Communications, Control and Computing, 1979, pp. 480–489.

    Google Scholar 

  40. J.F. Toland, On subdifferential calculus and duality in nonconvex optimization, Bull. Soc. Math. France, Mémoire 60 (1979), pp. 177–183.

    MathSciNet  MATH  Google Scholar 

  41. H. Tuy, A general deterministic approach to global optimization via d.c. programming, Fermat Days 1985: Mathematics for Optimization, North-Holland, Amsterdam, (1986), pp. 137–162.

    Google Scholar 

  42. H. Tuy, Introduction to Global Optimization, Les Cahiers du GERAD, Groupe d’Etudes et de Recherche en Analyse des Décision, Montréal, Québec, 1994.

    Google Scholar 

  43. H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, 1998.

    Book  MATH  Google Scholar 

  44. R. Varga Matrix iterative analysis, Prentice Hall, 1962.

    Google Scholar 

  45. Z. Zou, Richard.H. Bird, & Robert B. Schnabel, A Stochastic/Pertubation Global Optimization Algorithm for Distance Geometry Problems, J. of Global Optimization, 11(1997), pp. 91–105.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Modelling and Applied Optimization Group, Laboratory of Mathematics, CNRS UPRES A 60 85, National Institute for Applied Sciences-Rouen, BP 8, F 76 131, Mont Saint Aignan Cedex, France

    Thi Hoai An Le & Dinh Tao Pham

Authors
  1. Thi Hoai An Le
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  2. Dinh Tao Pham
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Editor information

Editors and Affiliations

  1. Princeton University, USA

    C. A. Floudas

  2. University of Florida, USA

    P. M. Pardalos

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© 2000 Springer Science+Business Media Dordrecht

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Le, T.H.A., Pham, D.T. (2000). D.C. Programming Approach for Large-Scale Molecular Optimization via the General Distance Geometry Problem. In: Floudas, C.A., Pardalos, P.M. (eds) Optimization in Computational Chemistry and Molecular Biology. Nonconvex Optimization and Its Applications, vol 40. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3218-4_18

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  • DOI: https://doi.org/10.1007/978-1-4757-3218-4_18

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4826-7

  • Online ISBN: 978-1-4757-3218-4

  • eBook Packages: Springer Book Archive

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