Deformation Methods of Global Optimization in Chemistry and Physics

  • Lucjan Piela
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 62)


Deformation of the target function in global optimization has been a novel possibility for the last decade. The techniques based on the deformation turned out to be related to a variety of fundamental laws: diffusion equation, time-dependent Schrödinger equations, Smoluchowski dynamics, Bloch equation of canonical ensemble evolution with temperature, Gibbs free-energy principle. The progress indicator of global optimization in those methods takes different physical meanings: time, imaginary time or the inverse absolute temperature. Despite of the fact that the phenomena described are different, the resulting global optimization procedures have a remarkable similarity. In the case of the Gaussian Ansatz for the wave function or density distribution, the underlying differential equations of motion for the Gaussian position and width are of the same kind for all the phenomena. The original potential energy function is smoothed by a convolution with a Gaussian distribution, its center denoting the current position in space during the minimization. The Gaussian position moves according to the negative gradient of the smoothed potential energy function. The Gaussian width is position dependent through the curvature of the smoothed potential energy function, and evolves according to the following rule. For sufficiently positive curvatures (close to minima of the smoothed potential) the width decreases, thus leading to a smoothed potential approaching the original potential energy function, while for negative curvatures (close to maxima) the width increases leading eventually to disappearance of humps of the original potential energy function. This allows for crossing barriers separating the energy basins. Some methods result in an additional term, which increases the width, when the potential becomes flat. This may be described as a feature allowing hunting for distant minima. Some deformation methods that are of non-convolutional character are also discussed.


Global Optimization Target Function Potential Energy Function Bloch Equation Deformation Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amara, P., Hsu, D., and Straub, J.E. (1993). Global energy minimum searches using an approximate solution of the imaginary time Schrödinger equation. The Journal of Physical Chemistry, 97: 6715.CrossRefGoogle Scholar
  2. Andricioaei, I. and Straub, J.E. (1998). Global optimization using bad derivatives: Derivative-free method for molecular energy minimization. Journal of Computational Chemistry, 19: 1445–1455.CrossRefGoogle Scholar
  3. Chandrasekharan, R. and Ramachandran, G.N. (1970). Studies on the conformation of amino acids. XI. Analysis of the observed side chain group conformation in proteins. International Journal of Peptide and Protein Research, 2: 223.Google Scholar
  4. Dixon, L.C.W. and Szegö, G.P. (1978). The global optimisation problem: an introduction. In Dixon, L.C.W. and Szegö, G.P., editors, Towards Global Optimisation 2, pages 1–15, Amsterdam, The Netherlands. North-Holland.Google Scholar
  5. Doye, J.P.K., Miller, M.A., and Wales, D.J. (1999). The double-funnel energy landscape of the 38-atom Lennard-Jones cluster. The Journal of Chemical Physics, 110 (14): 6896–6906.CrossRefGoogle Scholar
  6. Doye, J.P.K. and Wales, D.J. (1995). Calculation of thermodynamic properties of small Lennard-Jones clusters incorporating anharmonicity. The Journal of Chemical Physics, 102 (24): 9659–9672.CrossRefGoogle Scholar
  7. Elber, R. and Karplus, M. (1990). Enhanced sampling in molecular dynamics: Use of the time-dependent Hartree approximation for a simulation of carbon monoxide diffusion through Myoglobin. Journal of the American Chemical Society, 112: 9161.CrossRefGoogle Scholar
  8. Frauenfelder, H., Sligar, S.G., and Wolynes, P.G. (1991). The energy landscapes and motions of proteins. Science, 254: 1598.CrossRefGoogle Scholar
  9. Gordon, H.L. and Somorjai, R.L. (1992). Applicability of the method of smoothed functionals as a global minimizer for model polypeptides. The Journal of Physical Chemistry, 96: 7116.CrossRefGoogle Scholar
  10. Hoare, M.R. (1979). Structure and dynamics of simple microclusters. In Prigogine, I. and Rice, S.A., editors, Advances In Chemical Physics, volume 40, page 49. Wiley, New York, New York.CrossRefGoogle Scholar
  11. Hoare, M.R. and McInnes, J. (1983). Morphology and statistical statics of simple macroclusters. Advances in Physics, 32: 791.MathSciNetCrossRefGoogle Scholar
  12. Huber, T., Torda, A.E., and van Gunsteren, W.F. (1997). Structure optimization combining self-core interaction functions, the diffusion equation method and molecular dynamics. The Journal of Physical Chemistry, A101: 5926.CrossRefGoogle Scholar
  13. Huber, T. and van Gunsteren, W.F. (1998). SWARM-MD: Searching conformational space by cooperative molecular dynamics. The Journal of Physical Chemistry, 102: 5937.CrossRefGoogle Scholar
  14. Kirkpatrick, S., Gelatt Jr., C.D., and Vecchi, M.P. (1983). Optimization by simulated annealing. Science, 220: 671–680.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Kostrowicki, J. and Piela, L. (1991). Diffusion equation method of global minimization: Performance for the standard test functions. Journal of Optimization Theory and Applications, 69: 269.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Kostrowicki, J., Piela, L., Cherayil, B.J., and Scheraga, H.A. (1991). Performance of the diffusion equation method in searches for optimum structures of clusters of Lennard-Jones atoms. The Journal of Physical Chemistry, 95: 4113.CrossRefGoogle Scholar
  17. Kostrowicki, J. and Scheraga, H.A. (1992). Application of the diffusion equation method for the global optimization to oligopeptides. The Journal of Physical Chemistry, 96: 7442.CrossRefGoogle Scholar
  18. Landau, L.D. and Lifshitz, E.M. (1958). Quantum Mechanics,chapter VII. Pergamon Press, New York, New York.Google Scholar
  19. Liu, Z. and Berne, B.J. (1993). Method for accelerating chain folding and mixing. The Journal of Chemical Physics, 99: 6071.CrossRefGoogle Scholar
  20. Liwo, A., Pincus, M.R., Wawak, R.J., Rackovsky, S., and Scheraga, H.A. (1993). Prediction of protein conformation on the basis of a search for compact structures: Test on avian pancreatic polypeptide. Protein Science, 2: 1715.CrossRefGoogle Scholar
  21. Ma, J., Hsu, D., and Straub, J.E. (1993). Approximate solution of the classical Liouville equation using Gaussian phase packet dynamics: Application to enhanced equilibrium averaging and global optimization. The Journal of Chemical Physics, 99 (5): 4024–4035.CrossRefGoogle Scholar
  22. Ma, J. and Straub, J.E. (1994). Simulated annealing using classical density distribution. The Journal of Chemical Physics, 101: 533.CrossRefGoogle Scholar
  23. Mézard, M. and Visaroro, M.A. (1985). The microstructure of ultrametricity. Journal de Physique, 46: 1293.CrossRefGoogle Scholar
  24. Moré, J.J. and Wu, Z. (1996). Smoothing techniques for macromolecular global optimization. In Di Pillo, G. and Giannessi, F., editors, Nonlinear Optimization and Applications, pages 297–312. Plenum Press.Google Scholar
  25. Moré, J.J. and Wu, Z. (1997a). Global continuation for distance geometry problems. SIAM Journal on Optimization, 7: 814–836.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Moré, J.J. and Wu, Z. (1997b). Issues in large-scale global molecular optimization. In Biegler, L.T., Coleman, T., Conn, A.R., and Santosa, F.N., editors, Large Scale Optimization with Applications: Molecular Structure and Optimization, pages 99–122. Springer Verlag. Series IMA Volumes in Applied Mathematics and Applications, 94.Google Scholar
  27. More, J.J. and Wu, Z. (1999). Distance geometry optimization for protein structures. Journal of Global Optimization, 15: 219–234.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Nakamura, S., Ikeguchi, H. Hirose M.and, and Doi, J. (1995). Conformational energy minimization using a two-stage model. The Journal of Physical Chemistry, 99: 8374.Google Scholar
  29. Northby, J.A. (1987). Structure and binding of the Lennard-Jones clusters: 13 -n - 147. The Journal of Chemical Physics, 87: 6166.CrossRefGoogle Scholar
  30. Olszewski, J. Pillardy K.A. and Piela, L. (1992). Theoretically predicted lowest-energy structures of water clusters. Journal of Molecular Structure, 270: 277.CrossRefGoogle Scholar
  31. Olszewski, K., Piela, L., and Scheraga, H.A. (1992). Mean field theory as a tool for intramolecular conformational optimization. 1. Tests on terminally-blocked alanine and met-enkephalin. The Journal of Physical Chemistry, 96: 4672.CrossRefGoogle Scholar
  32. Olszewski, K.A., Piela, L., and Scheraga, H.A. (1993a). Mean field theory as a tool for intramolecular conformational optimization. 2. Tests on the homopolypeptides Decaglycine and Icosalanine. The Journal of Physical Chemistry, 97: 260.CrossRefGoogle Scholar
  33. Olszewski, K.A., Piela, L., and Scheraga, H.A. (1993b). Mean field theory as a tool for intramolecular conformational optimization. 3. Test on mellitin. The Journal of Physical Chemistry, 97: 267.CrossRefGoogle Scholar
  34. Piela, L. (1998). Search for the most stable structures on potential energy surfaces. Collection of Czechoslovak Chemical Communications, 63: 1368.CrossRefGoogle Scholar
  35. Piela, L., Kostrowicki, J., and Scheraga, H.A. (1989). The multiple-minima problem in the conformational analysis of molecules. Deformation of the potential energy hypersurface by the diffusion equation method. The Journal of Physical Chemistry, 93: 3339.CrossRefGoogle Scholar
  36. Piela, L., Olszewski, K.A., and Pillardy, J. (1994). On the stability of conformers. Journal of Molecular Structure, 308: 229.CrossRefGoogle Scholar
  37. Pillardy, J., Olszewski, K.A., and Piela, L. (1992). Performance of the shift method of global minimization in searches for optimum structures of clusters of Lennard-Jones atoms. The Journal of Physical Chemistry, 96: 4337.CrossRefGoogle Scholar
  38. Pillardy, J. and Piela, L. (1995). Molecular dynamics on deformed energy hypersurfaces. The Journal of Physical Chemistry, 99: 1 1805.Google Scholar
  39. Pillardy, J. and Piela, L. (1997). Smoothing techniques of global optimization: Distance scaling method in searches for most stable LennardJones atomic clusters. Journal of Computational Chemistry, 18: 2040.CrossRefGoogle Scholar
  40. Pillardy, J. and Piela, L. (1998). Multiple elliptical-Gaussian-density annealing as a tool for finding the most stable structures. Application to Lennard-Jones atomic clusters. Polish Journal of Chemistry, 72: 1849.Google Scholar
  41. Roitberg, A. and Elber, R. (1991). Modeling side chains in peptides and proteins: Application of the locally enhanced sampling and the simulated annealing methods to find minimum energy conformations. The Journal of Chemical Physics, 95 (12): 9277–9287.CrossRefGoogle Scholar
  42. Sali, A., Shakanovich, E.I., and Karplus, M. (1994). How does a protein fold? Nature, 369: 248.CrossRefGoogle Scholar
  43. Schelstraete, S., Schepens, W., and Verschelde, H. (1999). Energy minimization by smoothing techniques: a survey. In Balbuena, P.B. and Seminario, J.M., editors, Molecular Dynamics. From Classical to Quantum Methods, pages 129–185. Elsevier. Theor. Comput. Chem., Vol 7.Google Scholar
  44. Schelstraete, S. and Verschelde, H. (1997). Finding minimum-energy configurations of Lennard-Jones clusters using an effective potential. The The Journal of Physical Chemistry, 101: 315.Google Scholar
  45. Schütte, Ch. (1995). Smoothed molecular dynamics for thermally embedded systems. Report S.C. 95–14, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, Germany.Google Scholar
  46. Shakanovich, E.I. and Gutin, A.M. (1993). Engineering of stable and fast-folding sequences of model proteins. Proceedings of the National Academy of Sciences of the USA, 90: 7195.CrossRefGoogle Scholar
  47. Shalloway, D. (1992a). Application of the renormalization group to deterministic global minimization of molecular conformation energy functions. Journal of Global Optimization, 2: 281.MathSciNetzbMATHCrossRefGoogle Scholar
  48. Shalloway, D. (1992b). Packet annealing: A deterministic method for global minimization. Application to molecular conformation. In Floudas, C. and Pardalos, P., editors, Recent Advances in Global Optimization, page 433, Princeton, New Jersey. Princeton University Press.Google Scholar
  49. Skolnick, J. and Kolinski, A. (1990). Simulations of the folding of a globular protein. Science, 250: 1121.CrossRefGoogle Scholar
  50. Smoluchowski, M. (1916a). Drei Vorträge uber Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physikalische Zeitschrift, XVII: 557–571.Google Scholar
  51. Smoluchowski, M. (1916b). Drei Vorträge uber Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen (Schluss). Physikalische Zeitschrift, XVII: 585–599.Google Scholar
  52. Somorjai, R.L. (1991a). Novel approach for computing the global minimum of proteins. 1. General concepts, methods, and approximations. The Journal of Physical Chemistry, 95: 4141.CrossRefGoogle Scholar
  53. Somorjai, R.L. (1991b). Novel approach for computing the global minimum of proteins. 2. One-dimensional test cases. The Journal of Physical Chemistry, 95: 4147.CrossRefGoogle Scholar
  54. Stillinger, F.H. (1985). Role of potential-energy scaling in the low-temperature relaxation behavior of amorphous materials. Physical Review B, 32: 3134–3141.CrossRefGoogle Scholar
  55. Stillinger, F.H. and Stillinger, D.K. (1990). Cluster optimization simplified by interaction modification. The Journal of Chemical Physics, 93: 6106.CrossRefGoogle Scholar
  56. Straub, J.E. (1996). Optimization techniques with applications to proteins. In Elber, R., editor, Recent developments in Theoretical Studies of Proteins, page 137. World Scientific, Singapore.CrossRefGoogle Scholar
  57. Straub, J.E., Ma, J., and Amara, P. (1995). Simulated annealing using coarse grained classical dynamics: Smoluchowski dynamics in the Gaussian density approximation. The Journal of Chemical Physics, 103 (4): 1574–1581.CrossRefGoogle Scholar
  58. Thom, R. (1975). Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Benjamin-Cummings Publishing, Reading, Massachusetts.Google Scholar
  59. Verschelde, H., Schelstraete, S., Vandekerckhove, J., and Verschelde, J.L. (1997). An effective potential for calculating free energies. I. General concepts and approximations. The Journal of Chemical Physics, 106: 1556.CrossRefGoogle Scholar
  60. Wenzel, W. and Hamacher, K. (1999). Stochastic tunneling approach for global minimization of complex potential energy landscapes. Physical Review Letters, 82: 3003.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Wille, L.T. (2000a). Lennard-Jones clusters and the multiple-minima problem. In Stauffer, D., editor, Annual Reviews of Computational Physics VII, pages 25–60. World Scientific, Singapore.CrossRefGoogle Scholar
  62. Wille, L.T. (2000b). Simulated annealing and the topology of the potential energy surface of Lennard-Jones clusters. Comp. Mat. Sci., 17: 551.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Lucjan Piela
    • 1
  1. 1.Quantum Chemistry Laboratory Department of ChemistryUniversity of WarsawWarsawPoland

Personalised recommendations