Mathematical Programming

, 36:145

Symmetric minimum-norm updates for use in gibbs free energy calculations

  • Douglas E. Salane


This paper examines a type of symmetric quasi-Newton update for use in nonlinear optimization algorithms. The updates presented here impose additional properties on the Hessian approximations that do not result if the usual quasi-Newton updating schemes are applied to certain Gibbs free energy minimization problems. The updates derived in this paper are symmetric matrices that satisfy a given matrix equation and are least squares solutions to the secant equation. A general representation for this class of updates is given. The update in this class that has the minimum weighted Frobenius norm is also presented.

Key words

Gibbs free energy Symmetric updates nonlinear optimization 


  1. [1]
    C.G. Broyden, “The convergence of a class of double-rank minimization algorithms,”Journal of the Institute of Mathematics and its Applications 6 (1971) 76–90.MathSciNetGoogle Scholar
  2. [2]
    R.L. Crane, K.E. Hillstrom and M. Minkoff, “Solution of the general nonlinear programming problem with the subroutine VMCON,” Technical Report ANL-80-64, Argonne National Laboratory (Argonne, IL, 1980).Google Scholar
  3. [3]
    J.E. Dennis, Jr., and J.J. Moré, “Quasi-Newton methods, motivation and theory,”SIAM Review 19 (1977) 46–89.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J.E. Dennis, Jr., and R.B. Schnabel, “Least change secant updates for quasi-Newton methods,”SIAM Review 21 (1979) 443–459.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. Fletcher, “A new approach to variable metric algorithms,”Computer Journal 13 (1970) 317–322.MATHCrossRefGoogle Scholar
  6. [6]
    P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, New York, 1981).MATHGoogle Scholar
  7. [7]
    D. Goldfarb, “A family of variable metric methods derived by variational means,”Mathematics of Computation 24, (1970) 23–26.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J.P. Greenberg, J. H. Weare and C. H. Harvie, “An equilibrium computation algorithm for complex highly nonideal systems: application to silicate phase equilibria,”High Temperature Science, to appear.Google Scholar
  9. [9]
    G.P. Herring, “A note on generalized interpolation and the pseudo-inverse,”SIAM Journal on Numerical Analysis, 4 (1967).Google Scholar
  10. [10]
    A. Lucia and S. Macchietto, “New approach to the approximation of quantities involving physical properties derivatives in equation oriented process design,”American Institute of Chemical Engineers Journal 29 (1983) 205–216.Google Scholar
  11. [11]
    A. Lucia, D.C. Miller and A. Kumar, “Thermodynamically consistent quasi-Newton formulae,” contributed paper, 77th Annual American Institute of Chemical Engineers Meeting (San Francisco, CA) November 1984, to appear inAmerican Institute of Chemical Engineers Journal.Google Scholar
  12. [12]
    M.J.D. Powell, “A new algorithm for unconstrained optimization” in: J.B. Rosen, O.L. Mangasarian and K. Ritter, eds.,Nonlinear Programming (Academic Press, New York, 1970), 31–66.Google Scholar
  13. [13]
    M.J.D. Powell, “A fast algorithm for nonlinear constrained optimization calculations,” in: G.A. Watson, ed.,Numerical Analysis Proceedings, Biennial Conference Dundee 1977 (Springer-Verlag, Lecture notes in Mathematics 630, Berlin, 1978), 144–157.Google Scholar
  14. [14]
    R. Penrose, “A generalized inverse for matrices,”Proceedings of the Cambridge Philosophical Society 51 (1957) 406–413.MathSciNetCrossRefGoogle Scholar
  15. [15]
    D.E. Salane and R.P. Tewarson, “A unified derivation of symmetric quasi-Newton update formulas,”Journal of the Institute of Mathematics and its Applications 25 (1980) 29–36.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    D.E. Salane and R.P. Tewarson, “On symmetric minimum norm updates,”IMA Journal of Numerical Mathematics 1 (1981) 235–240.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    D.F. Shanno, “Conditioning of quasi-Newton methods for function minimization,”Mathematics of Computation 24 (1970) 647–656.CrossRefMathSciNetGoogle Scholar
  18. [18]
    W.R. Smith and R.W. Missen,Chemical Reaction Equilibrium Analysis Theory and Algorithms (John Wiley and Sons, New York, 1982).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

  • Douglas E. Salane
    • 1
  1. 1.Sandia National LaboratoriesAlbuquerqueUSA

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