Large-scale molecular optimization from distance matrices by a d.c. optimization approach

SIAM Journal on Optimization

Volumn 14, Issue 1, 2004, Pages 77-114

  An, Le Thi Hoai ^a   Tao, Pham Dinh ^a  

^a UNIVERSITÉ ET INSA DE ROUEN   (France)

Author keywords

d.c. algorithm (DCA); d.c. programming; Dissimilarity geometry problem; Distance geometry problem; Lagrangian duality; Reformulations; Subdifferential calculations

Indexed keywords

EID: 1542380919     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623498342794     Document Type: Article

References (41)

1. A.Y. ALFAKIH, A. KHANDANI, AND H. WOLKOWICZ, Solving Euclidean distance matrix completion problems via semidefinite programming. Computational optimization - A tribute to Olvi Mangasarian, Part I, Comput. Optim. Appl., 12 (1999), pp. 13-30.
2. H.M. BERMAN, J. WESTBROOK, Z. FENG, G. GILLILAND, T.N. BHAT, W. WEISSIG, I.N. SHINDYALOV, AND P.E. BOURNE, The Protein Data Bank, Nucleic Acids Res., 28 (2000), pp. 235-242; available online at http://www.resb.org/pdb/.
3. L.M. BLUMENTHAL, Theory and Applications of Distance Geometry, Oxford University Press, London, 1953.
4. G.M. CRIPPEN, Rapid calculation of coordinates from distance measures, J. Comput. Phys., 26 (1978), pp. 449-452.
5. G.M. CRIPPEN AND T.F. HAVEL, Distance Geometry and Molecular Conformation, John Wiley & Sons, New York, 1988.
6. V.F. DEMYANOV AND L.V. VASILEV, Nondifferentiable Optimization, Optimization Software Publications, New York, 1985.
7. J. DE LEEUW, Applications of convex analysis to multidimensional scaling, in Recent Developments in Statistics, J.R. Barra, F. Brodeau, G. Romier, and B. van Cutsem, eds., North-Holland, Amsterdam, 1977, pp. 133-145.
8. J. DE LEEUW, Convergence of the majorization method for multidimensional scaling, J. Classification, 5 (1988), pp. 163-180.
9. W. GLUNT, T.L. HAYDEN, AND M. RAYDAN, Molecular conformation from distance matrices, J. Comput. Chem., 14 (1993), pp. 114-120.
10. L. GUTTMAN, A general nonmetric technique for finding the smallest coordinate space for a configuration of point, Psychometrika, 33 (1968), pp. 469-506.
11. 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.
12. B.A. HENDRICKSON, The Molecule Problem: Determining Conformation from Pairwise Distances, Ph.D. thesis, Cornell University, Ithaca, NY, 1991.
13. B. HENDRICKSON, The molecule problem: Exploiting structure in global optimization, SIAM J. Optim., 5 (1995), pp. 835-857.
14. P.J. LAURENT, Approximation et Optimisation, Hermann, Paris, 1972.
15. M. LAURENT, Cuts, matrix completions and a graph rigidity, Math. Programming, 79 (1997), pp. 255-283.
16. H.A. LE THI, Contribution à l'optimisation non convexe et l'optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des Recherches, Université de Rouen, Rouen, France, 1997.
17. H.A. LE THI AND T. PHAM DINH, Solving a class of linearly constrained indefinite quadratic problems by d.c. algorithms, J. Global Optim., 11 (1997), pp. 253-285.
18. H.A. LE THI, T. PHAM DINH, AND L.D. MUU, Numerical solution for optimization over the efficient set by d.c. optimization algorithm, Oper. Res. Lett., 19 (1996), pp. 117-128.
19. H.A. LE THI AND T. PHAM DINH, The DC programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res., to appear.
20. P. MAHEY AND T. PHAM DINH, Partial regularization of the sum of two maximal monotone operators, M2AN Math. Model. Numer. Anal., 27 (1993), pp. 375-395.
21. P. MAHEY, S. OUALIBOUCH, AND T. PHAM DINH, Proximal decomposition on the graph of a maximal monotone operator, SIAM J. Optim., 5 (1995), pp. 454-466.
22. J.J. MORÉ AND Z. WU, Global continuation for distance geometry problems, SIAM J. Optim., 7 (1997), pp. 814-836.
23. J.J. MORÉ AND Z. WU, Issues in Large-Scale Molecular Optimization, Preprint MCS-P539-1095, Argonne National Laboratory, Argonne, IL, 1996.
24. J.J. MORÉ AND Z. WU, Distance Geometry Optimization for Protein Structures, Preprint MCS-P628-1296, Argonne National Laboratory, Argonne, IL, 1996.
25. A.M. OSTROWSKI, Solutions of Equations and Systems of Equations, Academic Press, New York, 1966.
26. T. PHAM DINH, Contribution à la théorie de normes et ses applications à l'analyse numérique, Thèse de Doctorat d'Etat Es Science, Université Joseph Fourier-Grenoble, Grenoble, France, 1981.
27. T. PHAM DINH, Convergence of subgradient method for computing the bound norm of matrices, Linear Algebra Appl., 62 (1984), pp. 163-182.
28. T. PHAM DINH, Algorithmes de calcul d'une forme quadratique sur la boule unité de la norme maximum, Numer. Math., 45 (1985), pp. 377-440.
29. T. PHAM DINH AND S. EL BERNOUSSI, Algorithms for solving a class of non convex optimization problems. Methods of subgradients, in Fermat Days 85. Mathematics for Optimization (Tolouse, France, 1985), North-Holland Math. Stud. 129, Elsevier, North-Holland, Amsterdam, 1986, pp. 249-271.
30. T. PHAM DINH AND S. EL BERNOUSSI, Duality in d.c. (difference of convex functions) optimization. Subgradient methods, in Trends in Mathematical Optimization, Internat. Schriftenreihe Numer. Math. 84, Birkhäuser, Basel, 1988, pp. 277-293.
31. T. PHAM DINH AND H.A. LE THI, Stabilité de la dualité lagrangienne en optimisation d.c. (différence de deux fonctions convexes), C.R. Acad, Paris Sér. I Math., 318 (1994), pp. 379-384.
32. T. PHAM DINH AND H.A. LE THI, A d.c. optimization algorithm for solving the trust-region subproblem, SIAM J. Optim., 8 (1998), pp. 476-505.
33. T. PHAM DINH AND H.A. LE THI, Convex analysis approach to d.c. programming: Theory, algorithms and applications (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday), Acta Math. Vietnam., 22 (1997), pp. 289-355.
34. B. POLYAK, Introduction to Optimization, Optimization Software Publications, New York, 1987.
35. R.T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
36. R.T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
37. J.B. SAXE, Embeddability of weighted Graphs in k-space is strongly NP-hard, in Proceedings of the 17th Allerton Conference in Communication, Control, and Computing, Monticello, IL, 1979, pp. 480-489.
38. N.Z. SHOR, Minimization Methods for Nondifferentiable Functions, Springer-Verlag, New York, Berlin, 1979.
39. J.F. TOLAND, On subdifferential calculus and duality in nonconvex optimization, Bull. Soc. Math. France, 60 (1979), pp. 177-183.
40. R. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
41. Z. ZOU, R.H. BIRD, AND R.B. SCHNABEL, A stochastic/perturbation global optimization algorithm for distance geometry problems, J. Global Optim., 11 (1997), pp. 91-105.