Advertisement

Some algorithms for nonlinear optimization

Chapter
  • 99 Downloads
Part of the Chapman and Hall Mathematics Series book series (CHMS)

Abstract

Many algorithms have been proposed for computing the optimum (or optima) of a mathematical programming problem, but there is no universal method. The simplex method for linear programming (3.3) is a highly efficient algorithm; while the number of iterations required to reach an optimum varies widely from one problem to another, the average number of iterations, for problems with constraints Axb ∈ ∝m/+ with A an m x n matrix with n much greater than m, is of the order of 2m, much less than might be expected from the number of vertices of the constraint set. (This remark does not apply to programming restricted to integer values.) If a nonlinear programming problem can be arranged so as to be solvable by a modified simplex method, this is commonly the most efficient procedure. In particular, a problem which allows an adequate approximation by piece- wise linear functions, of not too many variables, may be computed as a separable programming problem (3.4). Also a problem with a quadratic objective and linear constraints can be solved by a modified simplex method (4.6 and 7.6).

Keywords

Optimal Control Problem Mathematical Program Penalty Function Nonlinear Programming Problem Unconstrained Minimization 
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. Abadie, J. (Ed.) (1967), Nonlinear Programming, North-Holland, Amsterdam.Google Scholar
  2. Abadie, J. (Ed.) (1970), Integer and Nonlinear Programming, North-Holland, Amsterdam.Google Scholar
  3. Beale, E.M.L. (1967), Numerical methods, Chapter 7 of Abadie (1967).Google Scholar
  4. Beale, E.M.L. (1970), Advanced algorithmic features for general mathematical programming systems, Chapter 4 of Abadie (1970).Google Scholar
  5. Bennett, J.M. (1966), An approach to some structured linear programming problems, Operations Research, 14, 636–645.CrossRefGoogle Scholar
  6. Box, M.J., Davies, D., and Swann, W.H. (1969), Non-linear Optimization Techniques, Oliver and Boyd, Edinburgh. (I.C.I. Mathematical and Statistical Techniques for Industry, Monograph No. 5.)Google Scholar
  7. Broyden, C.G. (1972), Quasi-Newton methods, Chapter 6 of Murray (1972).Google Scholar
  8. Dantzig, G.B., and Wolfe, P. (1960), A decomposition principle for linear programs, Operations Research, 8, 101–111.CrossRefGoogle Scholar
  9. Fiacco, A.V., and McCormick, G.P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York.Google Scholar
  10. Gill, P.E., and Murray, W. (Eds.) (1974), Numerical Methods for Constrained Optimization, Academic Press, London.Google Scholar
  11. Lootsma, F.A. (1969), Hessian matrices of penalty functions for solving constrained minimization problems, Philips Res. Reports, 24, 332–330.Google Scholar
  12. Luenberger, D.G. (1969), Optimization by Vector Space Methods, Wiley, New York.Google Scholar
  13. Luenberger, D.G. (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading.Google Scholar
  14. Mangasarian, O.L. (1974), Nonlinear Programming, Theory and Computation, Chapter 6 of Elmaghraby, S.E., and Moder, J.J. (Eds.), Handbook of Operations Research, Van Nostrand Reinhold, New York.Google Scholar
  15. Mayne, D.Q., and Polak, E. (1975), First-order strong variation algorithms for optimal control, J. Optim. Theor. Appl., 16, 277–301.CrossRefGoogle Scholar
  16. Miele, A. (1975), Recent advances in gradient algorithms for optimal control problems, J. Optim. Theor. Appl., 17, 361–430.CrossRefGoogle Scholar
  17. McCormick, G.P., and Pearson, J.D. (1969), Variable metric methods and unconstrained optimization, in Fletcher, R. (Ed.), Optimization, Academic Press, London and New York.Google Scholar
  18. Murray, W. (ed.) (1972), Numerical Methods for Unconstrained Optimization, Academic Press, London.Google Scholar
  19. Pietrzykowski, T. (1969), An exact potential method for constrained maximum, SIAMJ. Numer. Anal., 6, 299–304.CrossRefGoogle Scholar
  20. Polak, E. (1971), Computational Methods in Optimization, Academic Press, New York.Google Scholar
  21. Rockafellar, R.T. (1974), Augmented Lagrange multiplier functions and duality in non-convex programming, SIAM J. Control, 12, 268–287.CrossRefGoogle Scholar
  22. Rosen, J.B. (1963), in Graves, R.L., and Wolfe, P., Recent Advances in Mathematical Programming, McGraw-Hill, New York, 159–176.Google Scholar
  23. Rosen, J.B., and Ornes, J.C. (1963), Solution of nonlinear programming problems by partitioning, Management Science, 10, 160–173.CrossRefGoogle Scholar
  24. Teo, K.-L. (1977), Computational methods of optimal control problems with application to decision making of a business firm, University of New South Wales, Private communication.Google Scholar
  25. Teo, K.-L., Reid, D.W., and Boyd, I.E. (1977), Stochastic optimal control theory and its computational methods, University of New South Wales, Research Report.Google Scholar
  26. Wolfe, P. (1967), Chapter 6 of Beale (1967).Google Scholar

Copyright information

© B. D. Craven 1978

Authors and Affiliations

  1. 1.University of MelbourneAustralia

Personalised recommendations