Annals of Operations Research

, Volume 42, Issue 1, pp 85–117 | Cite as

Global optimization for molecular conformation problems

  • Costas D. Maranas
  • Christodoulos A. Floudas


A primal-relaxed dual global optimization algorithm is presented along with an extensive review for finding the global minimum energy configurations of microclusters composed by particles interacting with any type of two-body central forces. First, the original nonconvex expression for the total potential energy is transformed to the difference of two convex functions (DC transformation) via an eigenvalue analysis performed for each pair potential that constitutes the total potential energy function. Then, a decomposition strategy based on the GOP algorithm [1–4] is designed to provide tight upper and lower bounds on the global minimum through the solutions of a sequence of relaxed dual subproblems. A number of theoretical results are included which expedite the computational effort by exploiting the special mathematical structure of the problem. The proposed approach attainsε-convergence to the global minimum in a finite number of iterations. Based on this procedure global optimum solutions are generated for small Lennard-Jones and Morse (a=3) microclustersn≤7. For larger clusters (8≤N≤24 for Lennard-Jones and 8≤N≤30 for Morse), tight lower and upper bounds on the global solution are provided which serve as excellent initial points for local optimization approaches.


Global Optimization Global Minimum Global Optimum Solution Total Potential Energy Central Force 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    C.A. Floudas and V. Visweswaran, Comp. Chem. Eng. 14(1990)1397.CrossRefGoogle Scholar
  2. [2]
    V. Visweswaran and C.A. Floudas, Comp. Chem. Eng. 14(1990)1419.CrossRefGoogle Scholar
  3. [3]
    V. Visweswaran and C.A. Floudas,Proc. Process Systems Engineering, PSE '91 (1991) p. I.6.1.Google Scholar
  4. [4]
    C.A. Floudas and V. Visweswaran, J. Optim. Theory Appl. (1992).Google Scholar
  5. [5]
    M.R. Hoare, Adv. Chem. Phys. 40(1979)49.Google Scholar
  6. [6]
    M.D. Morse and R.E. Smalley, Phys. Chem. 88(1984)228.Google Scholar
  7. [7]
    R.S. Bowles, J.J. Kolstad, J.M. Calo and R.P. Andres, Surf. Sci. 106(1981)117.CrossRefGoogle Scholar
  8. [8]
    E. Kay, Z. Phys. D 3(1986)251; J. Mol. Struct. 157(1987)43.Google Scholar
  9. [9]
    A. Brenner, D.H. Farrar and R.J. Goudsmit,Metal Clusters (Wiley, New York, 1986).Google Scholar
  10. [10]
    H. Poppa, D. Moorhead and K. Heinemann, Thin Solid Films 128(1985)251.CrossRefGoogle Scholar
  11. [11]
    M.A. Duncan and D.H. Rouvray, Microclusters, Sci. Amer. (December 1989).Google Scholar
  12. [12]
    R. Pool, Science 24(1990)1186.Google Scholar
  13. [13]
    T.L. Beck, J. Jellinek and R.S. Berry, J. Chem. Phys. 87(1987)545.CrossRefGoogle Scholar
  14. [14]
    R.S. Berry, Sci. Amer. (1990).Google Scholar
  15. [15]
    M.R. Hoare and J. McInnes, Adv. Phys. 32(1983)791.Google Scholar
  16. [16]
    L.T. Wille and J. Vennik, J. Phys. A 18(1985)L419.Google Scholar
  17. [17]
    M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).Google Scholar
  18. [18]
    A.H. Boerdijk, Philips Res. Rep. 7(1953)303.Google Scholar
  19. [19]
    H.S.M. Coxeter,Introduction to Geometry (Wiley, New York, 1961) chap. 22.Google Scholar
  20. [20]
    L. Fejes-Toth,Regular Figures (Macmillan, New York, 1954) chap. 9.Google Scholar
  21. [21]
    J.H. Conway and N.J.A. Sloane,Sphere Packings, Lattices and Groups (Springer, 1988).Google Scholar
  22. [22]
    M.R. Hoare and P. Pal, Nature Phys. Sci. 230(1971)5.Google Scholar
  23. [23]
    M.R. Hoare and P. Pal, Adv. Phys. 20(1971)161.Google Scholar
  24. [24]
    M.R. Hoare and P. Pal, J. Cryst. Growth 17(1972)77.CrossRefGoogle Scholar
  25. [25]
    M.R. Hoare and J. McInnes, Faraday Discussions Chem. Soc. 61(1972)12.CrossRefGoogle Scholar
  26. [26]
    L.T. Wille, Nature 34(1986)46.CrossRefGoogle Scholar
  27. [27]
    L. Piela, J. Kostrowicki and H.A. Scheraga, J. Phys. Chem. 93(1989)3339.CrossRefGoogle Scholar
  28. [28]
    F.H. Stillinger and T.A. Weber, Adv. Chem. Phys. 52(1988)1429.Google Scholar
  29. [29]
    D. Beeman, J. Comp. Phys. 20(1976)130.CrossRefGoogle Scholar
  30. [30]
    D.J. Evans and G.P. Morris, Comput. Phys. Rep. 1(1984)297.CrossRefGoogle Scholar
  31. [31]
    L.D. Heidi, J. Jellinek and R.S. Berry, J. Chem. Phys. 86(1987)6456.CrossRefGoogle Scholar
  32. [32]
    J. Jellinek, T.L. Beck and R.S. Berry, J. Chem. Phys. 84(1986)2783.CrossRefGoogle Scholar
  33. [33]
    J.D. Honeycutt and H.C. Andersen, J. Phys. Chem. 91(1987)4950.CrossRefGoogle Scholar
  34. [34]
    E.E. Polymeropoulos and J. Brickmann, Chem. Phys. Lett. 96(1983)73.CrossRefGoogle Scholar
  35. [35]
    E.E. Polymeropoulos and J. Brickmann, Surf. Sci. 156(1985)563.CrossRefGoogle Scholar
  36. [36]
    I.L. Garzon, X.P. Long, R. Kawai and J.H. Weare, Chem. Phys. Lett. 158(1989)525.CrossRefGoogle Scholar
  37. [37]
    I.L. Garzon, X.P. Long, R. Kawai and J.H. Weare, Z. Phys. D — Atoms, Molecules and Clusters 12(1989)81.Google Scholar
  38. [38]
    S. Erkoc and S. Katircioglu, Chem. Phys. Lett. 147(1988)476.CrossRefGoogle Scholar
  39. [39]
    N. Metropolis, A.W. Rosenbluth, A.H. Teller and E. Telleri, J. Chem. Phys. 21(1953)1087.CrossRefGoogle Scholar
  40. [40]
    N.H. Tsai and F.F. Abraham, Surf. Sci. 77(1978)465.CrossRefGoogle Scholar
  41. [41]
    N. Quirke and P. Sheng, Chem. Phys. Lett. 110(1984)63.CrossRefGoogle Scholar
  42. [42]
    H.U. Bohmer and S.D. Peyerimhoff, Stability and structure of singly-charged argon clusters Agn+,n=3–27. A Monte Carlo simulation, Z. Phys. D — Atoms, Molecules and Clusters 11(1989)239.Google Scholar
  43. [43]
    D.L. Freeman and J.D. Doll, J. Chem. Phys. 82(1984)462.CrossRefGoogle Scholar
  44. [44]
    C.Y. Yang and G. Bambakidis,Transition Metals, Institute of Physics Conf. Series 39(1977).Google Scholar
  45. [45]
    T. Halicioglu and P.J. White, J. Vac. Sci. Technol. 17(1980)1213.CrossRefGoogle Scholar
  46. [46]
    S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Science 220(1983)671.Google Scholar
  47. [47]
    D. Vanderbild and S.G. Louie, J. Comput. Phys. 56(1984)259.CrossRefGoogle Scholar
  48. [48]
    L.T. Wille, Chem. Phys. Lett. 133(1987)405.CrossRefGoogle Scholar
  49. [49]
    P. Ballone and P. Milani, Phys. Rev. 42(1990)3905.Google Scholar
  50. [50]
    D. Hohl, R.O. Jones, R. Car and M. Parrinello, J. Chem Phys 89(1988)6823.CrossRefGoogle Scholar
  51. [51]
    I.M. Navon, F.B. Brown and H. Robertson, Comp. Chem. 14(1990)305.CrossRefGoogle Scholar
  52. [52]
    D.G. Vlachos, L.D. Schmidt and R. Aris, J. Chem. Phys. 96(1992)6880.CrossRefGoogle Scholar
  53. [53]
    D.G. Vlachos, L.D. Schmidt and R. Aris, J. Chem. Phys. 96(1992)6891.CrossRefGoogle Scholar
  54. [54]
    D. Shalloway, J. Global Optim. 3(1992)281.MathSciNetGoogle Scholar
  55. [55]
    C.A. Floudas and P.M. Pardalos,Recent Advances in Global Optimization, (Princeton University Press, 1991).Google Scholar
  56. [56]
    J. Farges, M.F. de Feraudy, B. Raoult and G. Torchet, J. Chem. Phys. 78(1983)5067.CrossRefGoogle Scholar
  57. [57]
    J. Farges, M.F. de Feraudy, B. Raoult and G. Torchet, Surf. Sci. 156(1985)370.CrossRefGoogle Scholar
  58. [58]
    J. Farges, M.F. de Feraudy, B. Raoult and G. Torchet, J. Chem. Phys. 84(1986)3491.CrossRefGoogle Scholar
  59. [59]
    J.A. Northby, J. Chem. Phys. 87(1987)6166.CrossRefGoogle Scholar
  60. [60]
    G.L. Xue, Army High Performance Computing Research Center Preprint, University of Minnesota (1992).Google Scholar
  61. [61]
    R.S. Maier, J.B. Rosen and G.L. Xue, Army High Performance Computing Research Center Preprint 92-031, University of Minnesota (1992).Google Scholar
  62. [62]
    G.L. Xue, Army High Performance Computing Research Center Preprint 92-047, University of Minnesota (1992).Google Scholar
  63. [63.
    B.A. Murtagh and M.A. Saunders,MINOS 5.0 User's Guide (Systems Optimization Laboratory, Department of Operations Research, Stanford University, 1987).Google Scholar
  64. [64]
    C.D. Maranas and C.A. Floudas, J. Chem. Phys. 97(1992)10.CrossRefGoogle Scholar
  65. [65).
    D.G. Luenberger,Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1984).Google Scholar
  66. [66]
    A.M. Geoffrion, Optim. Theory Appl. 10(1972)237.CrossRefGoogle Scholar
  67. [67]
    G.L. Xue, R.S. Maier and J.B. Rosen, Army High Performance Computing Research Center Preprint 91-127, University of Minnesota (1992).Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Costas D. Maranas
    • 1
  • Christodoulos A. Floudas
    • 1
  1. 1.Department of Chemical EngineeringPrinceton UniversityPrincetonUSA

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