Abstract
Global convergence properties are established for a quite general form of algorithms for solving nonlinearly constrained minimization problems. A useful feature of the methods considered is that they can be implemented easily either with or without using quadratic programming techniques. A particular implementation, designed to be both efficient and robust, is described in detail. Numerical results are presented and discussed.
Similar content being viewed by others
References
Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.
Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, London, England, 1969.
Fletcher, R.,An Ideal Penalty Function for Constrained Optimization, Journal of Institute of Mathematics and Applications, Vol. 15, pp. 319–340, 1975.
Biggs, M. C.,Constrained Minimization Using Recursive Quadratic Programming: Some Alternative Subproblem Formulations, Toward Global Optimization, Edited by L. C. W. Dixon and G. P. Szegö, North-Holland Publishing Company, Amsterdam, Holland, 1975.
Han, S. P.,A Globally Convergent Method for Nonlinear Programming, Journal of Optimization Theory and Applications, Vol. 22, pp. 297–309, 1977.
Powell, M. J. D.,A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, Proceedings of Dundee Biennial Conference on Numerical Analysis, Edited by G. A. Watson, Springer-Verlag, Berlin, Germany, 1978.
Chamberlain, R. M.,Some Examples of Cycling in Variable Metric Methods for Constrained Minimization, Cambridge University, Technical Report No. DAMTP 78/NA2, 1978.
Powell, M. J. D.,Constrained Optimization by a Variable Metric Method, Cambridge University, Technical Report No. DAMTP 77/NA6, 1977.
Bertsekas, D. P.,Nondifferentiable Optimization via Approximation, Mathematical programming Study 3, Edited by M. Balinski and P. Wolfe, North Holland Publishing Company, Amsterdam, Holland, 1975.
Kort, B. W., andBertsekas, D. P.,Combined Primal Dual and Penalty Methods for Convex Programming, SIAM Journal of Control and Optimization, Vol. 14, pp. 268–294, 1976.
Han, S. P.,Penalty Lagrangian Methods via a Quasi-Newton Approach, Cornell University, Department of Computer Science, Technical Report No. TR-75-252, 1976.
Bertsekas, D. P.,Multiplier Methods: A Survey, Automatica, Vol. 12, pp. 133–145, 1976.
Powell, M. J. D.,Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches, UK Atomic Energy Research Establishment, Technical Report No. CSS-15, 1975.
Broyden, C. G., Dennis, J. E., andMore, J. J.,On the Local and Superlinear Convergence of Quasi-Newton Methods, Journal of Institute of Mathematics and Applications, Vol. 12, pp. 223–245, 1973.
Han, S. P.,Dual Variable Metric Algorithms for Constrained Optimization, SIAM Journal of Control and Optimization, Vol. 15, pp. 546–565, 1977.
Rosenbrock, H. H.,An Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal, Vol. 3, pp. 175–184, 1960.
Colville, A. R.,A Comparative Study on Nonlinear Programming Codes, IBM Scientific Center, New York, Report No. 320-2949, 1968.
Fletcher, R., andJackson, M. P.,Minimization of a Quadratic Function of Many Variables Subject Only to Lower and Upper Bounds, Journal of Institute of Mathematics and Applications, Vol. 14, pp. 159–174, 1974.
Dembo, R. S.,A Set of Geometric Test Problems and Their Solutions, Mathematical Programming, Vol. 10, pp. 192–213, 1976.
Author information
Authors and Affiliations
Additional information
Communicated by M. R. Hestenes
This work was carried out by the first author under the support of the Science Research Council (UK), Grant Nos. B/RG/95124 and GR/A/44480, which are gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Coope, I.D., Fletcher, R. Some numerical experience with a globally convergent algorithm for nonlinearly constrained optimization. J Optim Theory Appl 32, 1–16 (1980). https://doi.org/10.1007/BF00934840
Issue Date:
DOI: https://doi.org/10.1007/BF00934840