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Optimization methods for computing global minima of nonconvex potential energy functions

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Abstract

The minimization of potential energy functions plays an important role in the determination of ground states or stable states of certain classes of molecular clusters and proteins. In this paper we introduce some of the most commonly used potential energy functions and discuss different optimization methods used in the minimization of nonconvex potential energy functions. A very complete bibliography is also given.

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References

  1. L.L. Boyer and J.Q. Broughton, Statics and Dynamics of Icosahedrally Twinned and Single-Crystal fcc Clusters,Physical Review B, Vol. 42(90), pp. 11461–11468.

  2. B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan and M. Karplus, CHARMM: A Program for Macromolecular Energy, Minimization, and Dynamics Calculations,Journal of Computational Chemistry, Vol. 4(1983), pp. 187–217.

    Google Scholar 

  3. J.P. Brunet, A. Edelman, J.P. Mesirov, An Optimal Hypercube Direct N-body Solver on the Connection Machine, in Proceedings ofIEEE/ACM Supercomputing'90, pp. 748–752, IEEE Computer Society Press 1990.

  4. R.H. Byrd, E. Eskow, R.B. Schnabel, and S.L. Smith, Parallel Global Optimization: Numerical Methods, Dynamic Scheduling Methods, and Application to Molecular Configuration,Technical Report CU-CS-553-91, University of Colorado at Boulder, Department of Computer Science, Boulder, CO., October 1991.

  5. R.H. Byrd, E. Eskow, R.B. Schnabel, Global Optimization Methods for Molecular Configuration Problems, Presented at theFourth SIAM Conference on Optimization, May 11–13, 1992, Chicago, IL.

  6. J. Carrier, L. Greengard and V. Rokhlin, A Fast Adaptive Multipole Algorithm for Particle Simulations,SIAM Journal on Scientific and Statistic Computing, Vol. 9(1988), pp. 669–686.

    Google Scholar 

  7. M. Carson and J. Hermans, The Molecular Dynamics Workshop Laboratory, inMolecular Dynamics and Protein Structure, J. Hermans (ed.), University of North Carolina: Chapel Hill (1985), pp. 165–166.

    Google Scholar 

  8. H. S. Chan and K. A. Dill, The Protein Folding Problem,Physics Today, February 1993, pp. 24–32.

  9. T.W. Clark and J.A. McCammon, Parallelization of a Molecular Dynamics Non-Bonded Force Algorithm for MIMD Architecture,Computers and Chemistry, Vol. 14(1990), pp. 219–224.

    Google Scholar 

  10. T. Coleman, D. Shalloway, and Z. Wu, Isotropic Effective Energy Simulated Annealing Searches for Low Energy Molecular Cluster States,Compututational Optimization and Applications, Vol. 2(1993) pp. 145–170.

    Google Scholar 

  11. T. Coleman, D. Shalloway, and Z. Wu, A Parallel Build-Up Algorithm for Global Energy Minimizations of Molecular Clusters using Effective Energy Simulated Annealing,Journal of Global Optimization, in this issue (1994).

  12. J.H. Conway and N.J.A. Sloane,Sphere Packings, Lattices and Groups, Springer-Verlag, 1988.

  13. J. Farges, M.F. De Feraudy, B. Raoult and G. Torchet, Noncrystalline Structure of Argon Clusters. I. Polyicosahedral Structure ofAr N Clusters, 20 < N< 50,Journal of Chemical Physics, Vol. 78(1983), pp. 5067–5080.

    Google Scholar 

  14. J. Farges, M.F. De Feraudy, B. Raoult and G. Torchet, Cluster Models Made of Double Icosahedron Units,Surface Science, Vol. 156(1985), pp. 370–378.

    Google Scholar 

  15. J. Farges, M.F. De Feraudy, B. Raoult and G. Torchet, Noncrystalline Structure of Argon Clusters. I. Polyicosahedral Structure ofAr N Clusters, 50 < N< 750,Journal of Chemical Physics, Vol. 84(1986), pp. 5067–5080.

    Google Scholar 

  16. D.M. Ferguson and D.J. Raber, A New Approach to Probing Conformational Space with Molecular Mechanics: Random Incremental Pulse Search,Journal of the American Chemical Society, Vol. 111(1989), pp. 4371–4378.

    Google Scholar 

  17. D.M. Ferguson, W.A. Glauser and D.J. Raber, Molecular Mechanics Conformational Analysis of Cyclononane Using the RIPS Method and Comparison with Quantum-Mechanical Calculations,Journal of Computational Chemistry, Vol. 110(1990), pp. 903–910.

    Google Scholar 

  18. D.M. Ferguson and D.J. Raber, Molecular Mechanics Calculations of Several Lanthanide Complexes: An Application of the Random Incremental Pulse Search,Journal of Computational Chemistry, Vol. 11(1990), pp. 1061–1071.

    Google Scholar 

  19. I.Z. Fisher,Statistical Theory of Liquids, University of Chicago Press, 1964.

  20. D.G. Garrett, K.D. Kastella, D.M. Ferguson, New Results on Protein Folding from Simulated Annealing, submitted toJournal of the American Chemistry Society, 1992.

  21. L. Greengard,The Rapid Evaluation of Potential Fields in Particle Systems, The MIT Press, Cambridge, Massachusetts, 1988.

    Google Scholar 

  22. L. Greengard and V. Rokhlin, A Fast Algorithm for Particle Simulations,Journal of Computational Physics, Vol. 73(1987), pp. 325–348.

    Google Scholar 

  23. L. Greengard and W.D. Gropp, A Parallel Version of the Fast Multipole Method,Computers and Mathematics with Applications, Vol. 20(1990), pp. 63–71.

    Google Scholar 

  24. W.F. van Gunsteren and H. J.C. Berendsen, Computer Simulation of Molecular Dynamics: Methodology, Applications, and Perspectives in Chemistry,Angew. Chem. Int. Ed. Engl., Vol. 29(1990), pp. 992–1023.

    Google Scholar 

  25. D.A. Hinds and M. Levitt, A Lattice Model for Protein Structure Prediction at Low Resolution,Proceedings of the National Academy of Sciences USA, Vol. 89(1992), pp. 2536–2540.

    Google Scholar 

  26. M.R. Hoare, Structure and Dynamics of Simple Microclusters,Advances in Chemical Physics, Vol. 40(1979), pp. 49–135.

    Google Scholar 

  27. D. Hohl, R. Idaszak, and R.O. Jones, Quantum Molecular Modeling with Simulated Annealing — A Distributed Processing and Visualization Application, in Proceedings ofACM/IEEE Svpercomputing'90, pp. 816–825, IEEE Computer Society Press 1990.

  28. J. Danna Honeycutt and Hans C. Andersen, Molecular Dynamics Study of Melting and Freezing of Small Lennard-Jones Clusters,Journal of Physical Chemistry, Vol. 91(1987), pp. 4950–4963.

    Google Scholar 

  29. R. Horst and H. Tuy,Global Optimization, Deterministic Approaches, 2nd Edition, Springer-Verlag, 1993.

  30. R.S. Judson, M.E. Colvin, J.C. Meza, A. Huffer, and D. Gutierrez, Do Intelligent Configuration Search Techniques Outperform Random Search for Large Molecules?,Sandia Report SAND91-8740, Sandia National Laboratories, Center for Computational Engineering, Livermore, CA., December 1991.

    Google Scholar 

  31. S. Kirkpatrick, C.D. Gelatt, Jr., M.P. Vecchi, Optimization by Simulated Annealing,Science, Vol. 220(1983), pp. 671–680.

    Google Scholar 

  32. S. Kirkpatrick, Optimization by Simulated Annealing: Quantitative Studies,Journal of Statistical Physics, Vol. 34(1984), pp. 975–986.

    Google Scholar 

  33. A. Kolinski, M. Milik, and J. Skolnick, Static and Dynamic Properties of a New Lattice Model of Polypeptide Chains,Journal of Statistical Physics, Vol. 94(1991), pp. 3978–3985.

    Google Scholar 

  34. J. Kostrowicki and L. Piela, Diffusion Equation Method of Global Minimization: Performance for Standard Test Functions,Journal of Optimization Theory and Applications, Vol. 69(1991), pp. 269–284.

    Google Scholar 

  35. J. Kostrowicki, L. Piela, B.J. Cherayil and H.A. Scheraga, Performance of the Diffusion Equation Method in Searches for Optimum Structures of Clusters of Lennard-Jones Atoms,Journal of Physical Chemistry, Vol. 95(1991), pp. 4113–4119.

    Google Scholar 

  36. J. Kostrowicki and H.A. Scheraga, Application of the Diffusion Equation Method for Global Optimization to Oligopeptides,Journal of Physical Chemistry, Vol. 96(1992), pp. 7442–7449.

    Google Scholar 

  37. S.M. Le Grand and K.M. Merz Jr., The Application of the Genetic Algorithm to the Minimization of potential Energy Functions,Journal of Global Optimization, Vol. 3(1993), pp. 49–66.

    Google Scholar 

  38. M. Levitt, Protein Folding by Restained Energy Minimization and Molecular Dynamics,Journal of Molecular Biology, Vol. 170(1983), pp. 723–764.

    Google Scholar 

  39. Z. Li and H. Scheraga, Monte Carlo-Minimization Approach to the Multiple-Minima Problem in Protein Folding,Proceedings of the National Academy of Sciences USA, Vol. 84(1987), pp. 6611–6615.

    Google Scholar 

  40. R.S. Maier, J.B. Rosen, G.L. Xue, A Discrete-Continuous Algorithm for Molecular Energy Minimization, in Proceedings ofIEEE/ACM Supercompuiing'91, pp. 778–786, IEEE Computer Society Press 1992.

  41. C.D. Maranas and C.A. Floudas, Global Optimization of Lennard-Jones Microclusters,Journal of Chemical Physics, 97, 10, 7667 (1992).

    Google Scholar 

  42. C.D. Maranas and C.A. Floudas, Global Optimization for Molecular Conformation Problems,Annals of Operations Research, Vol. 42(1993), pp. 85–117.

    Google Scholar 

  43. C.D. Maranas and C.A. Floudas, Global Minimum Potential Energy Conformations of Small Molecules,Journal of Global Optimization, in this issue (1994).

  44. A.I. Melcuk, R.C. Giles and H. Gould, Molecular Dynamics Simulation of Liquids on the Connection Machine,Computers in Physics, May/Jun 1991, pp. 311–318.

  45. N. Metropolis, A. Rosenbluth, A. Teller, E. Teller, Equation of Several State Calculations by Fast Computing Machines,Journal of Chemical Physics, Vol. 21(1953), pp. 1087–1892.

    Google Scholar 

  46. S.G. Nash, User's Guide for TN/TNBC: Fortran Routines for Nonlinear Optimization,Report 397, Mathematical Sciences Department, The Johns Hopkins University, 1984.

  47. S.G. Nash, Preconditioning of Truncated-Newton Methods,SIAM Journal on Scientific and Statistic Computing, Vol. 6(1985), pp. 599–616.

    Google Scholar 

  48. D. R. Nelson, T. Piran, and S. Weinberg (Editors), Statistical Mechanics of Membranes and Surfaces,World Scientific, 1989.

  49. J.A. Northby, Structure and Binding of Lennard-Jones Clusters: 13≤n ≤ 147,Journal of Chemical Physics, Vol. 87(1987), pp. 6166–6178.

    Google Scholar 

  50. K.A. Olszewsky, L. Piela and H.A. Scheraga, Mean Field Theory as a Tool for Inter-Molecular Conformational Optimization. I. Tests on Terminally Blocked Alanine and Met-Enkephalin,Journal of Physical Chemistry, Vol. 96(1992), pp. 4672–4676.

    Google Scholar 

  51. P.M. Pardalos and J.B. Rosen,Constrained Global Optimization, Algorithms and Applications, Springer-Verlag Lecture Notes in Computer Science 268, 1987.

  52. H.G. Petersen and J.W. Perram, Molecular Dynamics on Transputer Arrays. I. Algorithm Design, Programming Issues, Timing Experiments and Scaling Projections,Molecular Physics, Vol. 67(1989), pp. 849–860.

    Google Scholar 

  53. L. Piela, J. Kostrowicki and H.A. Scheraga, The Multiple-Minima Problem in the Conformational Analysis of Molecules. Deformation of the Potential Energy Hypersurface by the Diffusion Equation Method,Journal of Physical Chemistry, Vol. 93(1989), pp. 3339–3346.

    Google Scholar 

  54. B. Raoult, J. Farges, M.F. De Feraudy and G. Torchet, Stability of Relaxed Lennard-Jones Models Made of 500 to 6000 Atoms,Z. Phys. D — Atoms, Molecules and Clusters, Vol. 12(1989), pp. 85–87.

    Google Scholar 

  55. B. Raoult, J. Farges, M.F. De Feraudy and G. Torchet, Comparison between Icosahedral, Decahedral and Crystalline Lennard-Jones Models containing 500 to 6000 Atoms,Philosophical Magazine B, Vol. 60(1989), pp. 881–906.

    Google Scholar 

  56. A. Rey and J. Skolnick, Comparison of Lattice Monte Carlo Dynamics and Brownian Dynamics Folding Pathways of α-Helical Hairpins, Journal of Chemical Physics, Vol. 158(1991), pp. 199–219.

    Google Scholar 

  57. F.M. Richards, The Protein Folding Problem,Scientific American, January, 1991, pp. 54–63.

  58. D.R. Ripoll, S.J. Thomas, A Parallel Monte Carlo Search Algorithm for the Conformational Analysis of Proteins, in Proceedings ofIEEE/A CM Supercomputing '90, pp. 94–102. IEEE Computer Society Press 1990.

  59. T. Schlick and M. Overtoil, A Powerful Truncated Newton Method for Potential Energy Minimization,Journal of Computational Chemistry, Vol. 8(1987), pp. 1025–1039.

    Google Scholar 

  60. E. Shakhnovich, G. Farztdnov, A.M. Gutin and M. Karplus, Protein Folding Bottlenecks: A Lattice Monte Carlo Simulation,Physical Review Letters, Vol. 67(1991), pp. 1665–1669.

    Google Scholar 

  61. D. Shalloway, Packet Annealing: A Deterministic Method for Global Minimization. Application to Molecular Conformation, InRecent Advances in Global Optimization, C. Floudas and P. Pardalos (eds.), Princeton University Press: Princeton, N.J., (1992) pp. 433–477.

    Google Scholar 

  62. D. Shalloway, Application of the Renormalization Group to Deterministic Global Minimization of Molecular Conformation Energy Functions,Journal of Global Optimization, Vol. 2(1992), pp. 281–311.

    Google Scholar 

  63. R.I. Somorjai, Novel Approach for Computing the Global Minimum of Proteins. I. General Concepts, Methods and Approximations,Journal of Physical Chemistry, Vol. 95(1991), pp. 4141–4146.

    Google Scholar 

  64. F. Sullivan, R.D. Mountain and J. O'Connell, Molecular Dynamics on Vector Computers,Journal of Computational Physics, Vol. 61(1985), pp. 138–153.

    Google Scholar 

  65. P. Tamayo, J.P. Mesirov and B.M. Boghsian, Parallel Approaches to Short Range Molecular Dynamics Simulations, in Proceedings ofIEEE/ACM Supercomputing '91, pp. 462–470, IEEE Computer Society Press 1991.

  66. J.M. Troyer and F.E. Cohen, Simplified Models for Understanding and Predicting Protein Structure, inReviews in Computational Chemistry II, pp. 57–80, K.B. Lipkowitz and D.B. Boyd eds, VCH Publishers, 1991.

  67. S. Vajda and C. Delisi, Determining Minimum Energy Conformations of Polypeptides by Dynamic Programming,Biopolymers, Vol. 29(1990), pp. 1755–1772.

    Google Scholar 

  68. W.F. van Gunsteren, H.J.C. Berendsen, J. Hermans, W.G.J. Hol and J.P.M. Postma, Computer Simulation of the Dynamics of Hydrated Protein Crystals and Its Comparison with X-Ray Data,Proceedings of the National Academy of Sciences of the USA, Vol. 80(1983), pp. 4315–4319.

    Google Scholar 

  69. D.G. Vlachos, L.D. Schmidt, and R. Aris, Structures of Small Metal Clusters: Phase Transitions and Isomerization,AHPCRC Preprint 91-69, University of Minnesota, Minneapolis, MN 55415, 1991.

    Google Scholar 

  70. D.G. Vlachos, L.D. Schmidt, and R. Aris, Structures of Small Metal Clusters: Low Temperature Behavior,AHPCRC Preprint 91-70, University of Minnesota, Minneapolis, MN 55415, 1991.

    Google Scholar 

  71. Benjamin W. van de Waal, Stability of Face-Centered Cubic and Icosahedral Lennard-Jones Clusters,Journal of Chemical Physics, Vol. 90(1989), pp. 3407–3408.

    Google Scholar 

  72. L. T. Watson, R.T. Haftka, F.H. Lutze, R.H. Plautt, and P.Y. Shin, The Application of Globally Convergent Homotopy Methods to Nonlinear Optimization, InAdvances in Numerical Partial Differential Equations and Optimization, S. Gomez, J.P. Hennart, and R.A. Tapia (eds.), SIAM, Philadelphia, PA, 1991, pp. 284–298.

    Google Scholar 

  73. P.K. Weiner and P.A. Kollman, AMBER: Assisted Model Building with Energy Refinement. A General Program for Modeling Molecules and Their Interactions,Journal of Computational Chemistry, Vol. 2(1981), pp. 287–303.

    Google Scholar 

  74. L.T. Wille, Minimum-Energy Configurations of Atomic Clusters: New Results Obtained by Simulated Annealing,Chemical Physics Letters, Vol. 133(1987), pp. 405–410.

    Google Scholar 

  75. K.G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem.Reviews of Modern Physics., Vol. 47(1975), pp. 773–840.

    Google Scholar 

  76. S.R. Wilson and W. Cui, Applications of Simulated Annealing to Peptides,Biopolymers, Vol. 29(1990), pp. 225–235.

    Google Scholar 

  77. Z. Wu, The Effective Energy Transformation Scheme as a General Continuation Approach to Global Optimization with Application to Molecular Conformation,Technical Report CTC93TR143, Advanced Computing Research Institute, Cornell University, Ithaca, 1993.

    Google Scholar 

  78. G.L. Xue, R.S. Maier, J.B. Rosen, Minimizing the Lennard-Jones Potential Function on a Massively Parallel Computer, in Proceedings of1992 ACM International Conference on Supercomputing, pp. 409–416, ACM Press, 1992.

  79. G.L. Xue, Molecular Conformation on the CM-5 by Parallel Two-Level Simulated Annealing,Journal of Global Optimization, in this issue (1994).

  80. G.L. Xue, Improvement of the Northby Algorithm for Molecular Conformation: Better Results, accepted for publication inJournal of Global Optimization.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, 32611, Gainesville, FL

    Panos M. Pardalos

  2. Section of Biochemistry, Molecular and Cell Biology, Cornell University, 14853, Ithaca, NY

    David Shalloway

  3. Department of Computer Science and Electrical Engineering, University of Vermont, 05405, Burlington, VT

    Guoliang Xue

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  1. Panos M. Pardalos
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  2. David Shalloway
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  3. Guoliang Xue
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Also a researcher at the Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN 55415

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Pardalos, P.M., Shalloway, D. & Xue, G. Optimization methods for computing global minima of nonconvex potential energy functions. J Glob Optim 4, 117–133 (1994). https://doi.org/10.1007/BF01096719

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