Abstract
Most nonlinear programming problems consist of functions which are sums of unary functions of linear functions. Advantage can be taken of this form to calculate second and higher order derivatives easily and at little cost. Using these, high order optimization techniques such as Halley's method can be utilized to accelerate the rate of convergence to the solution. These higher order derivatives can also be used to compute second order sensitivity information. These techniques are applied to the solution of the classical chemical equilibrium problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Alefeld, “On the convergence of Halley's method,”American Mathematical Monthly 88 (1981) 530–536.
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer,Nonlinear Parametric Optimization (Akademie-Verlag, Berlin, 1982).
G. Emami, “Evaluating strategies for Newton's method using a numerically stable generalized inverse algorithm,” Dissertation, Department of Operations Research, George Washington University (Washington, DC, 1978).
A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).
A.V. Fiacco,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press, New York, 1983).
F. Ghotb, “Newton's method for linearly constrained optimization problems,” Dissertation, Department of Operations Research, George Washington University (Washington, DC, 1980).
E. Halley “Methodus nova, accurata & facilis inveiendi radices aequationum quarumque generaliter, sine praevia reductione,”Philosophical Transactions of the Royal Society of London 18 (1694) 136–148. [In Latin.]
R.H.F. Jackson, “Tensors, polyads, and high-order methods in factorable programming,” Dissertation,
R.H.F. Jackson and G.P. McCormick, “The polyadic structure of factorable function tensors with application to high-order minimization techniques,”Journal of Optimization Theory and Applications 51 (1986) 63–94.
R.H.F. Jackson and G.P. McCormick, “Second-order sensitivity analysis in factorable programming: theory and applications,”Mathematical Programming 41 (1988) 1–27.
H. Markowitz,Portfolio Selection (Wiley, New York, 1959).
A. Sofer, “Computationally efficient techniques for generalized inversion in nonlinear programming,” Dissertation, Department of Operations Research, George Washington University (Washington, DC, 1984).
B.G. White, S.H. Johnson and G.B. Dantzig, “Chemical equilibrium in complex mixtures,”Journal of Chemical Physics 28 (1958) 751–755.
Author information
Authors and Affiliations
Additional information
Supported by National Science Foundation grant ECS-8709795, co-funded by the U.S. Air Force Office of Scientific Research and by the Office of Naval Research grant N00014-86-K0052.
Supported by National Science Foundation grant ECS-8709795, co-funded by the U.S. Air Force Office of Scientific Research.
Rights and permissions
About this article
Cite this article
McCormick, G.P., Sofer, A. Optimization with unary functions. Mathematical Programming 52, 167–178 (1991). https://doi.org/10.1007/BF01582885
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582885