Advertisement

Potential Transformation Method for Global Optimization

  • Robert A. Donnelly
Chapter
  • 221 Downloads
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 94)

Abstract

Several techniques for global optimization treat the objective function f as a force-field potential. In the simplest case, trajectories of the differential equation \(m\ddot x = - \nabla f\) sample regions of low potential while retaining the energy tosurmount passes which might block the way to regions of even lower local minima. A potential transformation is an increasing function V: R → R. It determines a new potential g = V(f), with the same minimizers as f, and new trajectories satisfying \(m\ddot x = - \nabla g = - \frac{{dV}}{{df}}\nabla f.\)We discuss a class of potential transformations that greatly increase the attractiveness of low local minima. As a special case, this provides a new approach to an equation proposed by Griewank for global optimization. Practical methods for implementing these ideas are discussed, and the method is applied to three test problems.

Keywords

Simulated Annealing Global Optimization Global Optimization Algorithm Potential Transformation Steep Descent Direction 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Iupac-iub commission on biochemical nomenclature. Biochemistry, 9:3174–3179, 1970.Google Scholar
  2. [2]
    F. Alluffi-Pentini, V. Parisi, and F. Zirilli. Global optimization and stochastic differential equations. J. Optim. Theory Appl., 47:1–16, 1985.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. A. Barkerand D. Henderson. Dynamics of proteins and nucleic acids. Rev. Mod. Phys., 48:587–671, 1976.CrossRefGoogle Scholar
  4. [4]
    H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys., 81:3684–3690, 1984.CrossRefGoogle Scholar
  5. [5]
    R. A. P Butler and E. E. Slaminka. An evaluation of the sniffer global optimization algorithm using standard test functions. J. Comp. Phys., 99:28–32, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    R. Car and M. Parinello. Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett., 55:2471–2474, 1985.CrossRefGoogle Scholar
  7. [7]
    L. C. W. Dixon and G. P. Szego, editors. Towards Global Optimization, Volume 2. Elsevier, New York, 1978.Google Scholar
  8. [8]
    R. A. Donnelly and J. W. Rogers, Jr. A discrete search technique for global optimization. Intl. J. Quantum Chem., 22:507–513, 1988.CrossRefGoogle Scholar
  9. [9]
    A. O. Griewank. Generalized descent for global optimization. J. Optim. Theory Appl., 34:11–39, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    T.S. Harvey and A.J. Winkinson and I.D. Campbell. The solution structure of human transforming growth factor α. Enr. J. Biochem., 989:555–562, 1991.CrossRefGoogle Scholar
  11. [11]
    M.R. Hoare and J. Mclnnes. Morphology and statistical statics of simple micro-clusters. Adv. Phys., 32:791, 1983.CrossRefGoogle Scholar
  12. [12]
    M.R. Hoare and P. Pal. Adv. Phys., 20:161, 1971.CrossRefGoogle Scholar
  13. [13]
    M.R. Hoare and P. Pal. Adv. Phys., 24:645, 1975.CrossRefGoogle Scholar
  14. [14]
    S. Kirkpatrick, Jr. G. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220:671–680, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. A. McCammon and S. C. Harvey. Dynamics of Proteins and Nucleic Acids. Cambridge University Press, New York, 1987.CrossRefGoogle Scholar
  16. [16]
    J.A. Northby. Structure and binding of lennard-jones clusters. J. Chem. Phys., 87:6166, 1987.CrossRefGoogle Scholar
  17. [17]
    M.L. Papay. Glide program optimization results. Technical report, TRW Defense Systems Group, San Bernadino, CA, 1989. Unpublished.Google Scholar
  18. [18]
    R. Stephen, Berry Ralph, E. Kunz. Statistical interpretation of topographies adn dynamics of multidimensional potentials. J. Chem. Phys., 103:1904, 1995.CrossRefGoogle Scholar
  19. [19]
    J. W. Rogers, Jr and R. A. Donnelly. A search technique for global optimization in a chaotic environment. J. Optim. Theory Appl., 61:111–121, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [19]
    J. W. Rogers, Jr and R. A. Donnelly. Potential transformation methods for large-scale global optimization. SIAM J. Opt., 5:871, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.P. Ryckhaert, G. Ciccotti, and H. J. C. Berendsen. Time-dependent constraints. J. Corn-put. Phys., 23:327, 1977.CrossRefGoogle Scholar
  22. [22]
    Panos M. Pardalos, David Shalloway and Guoliang Xue. Optimization methods for computing global minima of nonconvex potential energy functions. J. Global Opt., 4:117, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Thomas Coleman, David Shalloway and Zhijun Wu. Build-up algorithms for global energy minimization of molecular clusters using effective energy simulated annealing. J. Global Opt., 4:171, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    E. E. Slaminka and K. D. Woerner. Central configurations and a theorem of palmore. Celestial Mech. Dyn. Astron., 48:347–355, 1990.MathSciNetCrossRefGoogle Scholar
  25. [25]
    J. A. Synman and L. P. Fatti. A multi-start global minimization algorithm with dynamic search trajectories. J. Optim. Theory Appl, 54:121–141, 1987.MathSciNetCrossRefGoogle Scholar
  26. [26]
    A. A. Torn. A search clustering approach to global optimization. In L. C. W. Dixon and G. P. Szego, editors, Towards Global Optimization, Volume 2, New York, 1978. Elsevier.Google Scholar
  27. [27]
    C.J. Tsai and K.D. Jordan. Use of the eigenmode method to locate the stationary points of the potential energy surfaces of selected argon and water clusters. J.Phys. CHem., 93:11227, 1993.CrossRefGoogle Scholar
  28. [28]
    David van der Spoel, Rudi van Drunen, and Herman J.C. Berendsen. Groningenmachine for simulating chemistry, gromax user manual. Technical report, BIOSON Research Institute, Nijenborgh 4, NL 9747 AG, Groningen, Netherlands,1994.Google Scholar
  29. [29]
    D. Vanderbuilt and S. G. Louie. Optimization by simulated annealing. J. Comput.Physics, 56:259, 1984.CrossRefGoogle Scholar
  30. [30]
    Guoliang Xue. Improvement of the northby algorithm for molecular conformation: Better solutions. J. Global Opt., 4:425, 1994.CrossRefzbMATHGoogle Scholar
  31. [31]
    Guoliang Xue. Molecular conformation on the cm-5 by parallel two-level simulated annealing. J. Global Opt., 4:187, 1994.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Robert A. Donnelly
    • 1
  1. 1.Chemistry DepartmentAuburn UniversityAuburnUSA

Personalised recommendations