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 Ax — b ∈ ∝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).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abadie, J. (Ed.) (1967), Nonlinear Programming, North-Holland, Amsterdam.
Abadie, J. (Ed.) (1970), Integer and Nonlinear Programming, North-Holland, Amsterdam.
Beale, E.M.L. (1967), Numerical methods, Chapter 7 of Abadie (1967).
Beale, E.M.L. (1970), Advanced algorithmic features for general mathematical programming systems, Chapter 4 of Abadie (1970).
Bennett, J.M. (1966), An approach to some structured linear programming problems, Operations Research, 14, 636–645.
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.)
Broyden, C.G. (1972), Quasi-Newton methods, Chapter 6 of Murray (1972).
Dantzig, G.B., and Wolfe, P. (1960), A decomposition principle for linear programs, Operations Research, 8, 101–111.
Fiacco, A.V., and McCormick, G.P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York.
Gill, P.E., and Murray, W. (Eds.) (1974), Numerical Methods for Constrained Optimization, Academic Press, London.
Lootsma, F.A. (1969), Hessian matrices of penalty functions for solving constrained minimization problems, Philips Res. Reports, 24, 332–330.
Luenberger, D.G. (1969), Optimization by Vector Space Methods, Wiley, New York.
Luenberger, D.G. (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley, Reading.
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.
Mayne, D.Q., and Polak, E. (1975), First-order strong variation algorithms for optimal control, J. Optim. Theor. Appl., 16, 277–301.
Miele, A. (1975), Recent advances in gradient algorithms for optimal control problems, J. Optim. Theor. Appl., 17, 361–430.
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.
Murray, W. (ed.) (1972), Numerical Methods for Unconstrained Optimization, Academic Press, London.
Pietrzykowski, T. (1969), An exact potential method for constrained maximum, SIAMJ. Numer. Anal., 6, 299–304.
Polak, E. (1971), Computational Methods in Optimization, Academic Press, New York.
Rockafellar, R.T. (1974), Augmented Lagrange multiplier functions and duality in non-convex programming, SIAM J. Control, 12, 268–287.
Rosen, J.B. (1963), in Graves, R.L., and Wolfe, P., Recent Advances in Mathematical Programming, McGraw-Hill, New York, 159–176.
Rosen, J.B., and Ornes, J.C. (1963), Solution of nonlinear programming problems by partitioning, Management Science, 10, 160–173.
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.
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.
Wolfe, P. (1967), Chapter 6 of Beale (1967).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1978 B. D. Craven
About this chapter
Cite this chapter
Craven, B.D. (1978). Some algorithms for nonlinear optimization. In: Mathematical Programming and Control Theory. Chapman and Hall Mathematics Series. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5796-1_7
Download citation
DOI: https://doi.org/10.1007/978-94-009-5796-1_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-412-15500-0
Online ISBN: 978-94-009-5796-1
eBook Packages: Springer Book Archive