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Rates of convergence for a method of centers algorithm

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Abstract

Convergence of a method of centers algorithm for solving nonlinear programming problems is considered. The algorithm is defined so that the subproblems that must be solved during its execution may be solved by finite-step procedures. Conditions are given under which the algorithm generates sequences of feasible points and constraint multiplier vectors that have accumulation points satisfying the Fritz John or the Kuhn-Tucker optimality conditions. Under stronger assumptions, linear convergence rates are established for the sequences of objective function, constraint function, feasible point, and multiplier values.

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References

  1. Huard, P.,Resolution of Mathematical Programming with Nonlinear Constraints by the Method of Centers, Nonlinear Programming, Edited by J. Abadie, North-Holland Publishing Company, Amsterdam, Holland, 1967.

    Google Scholar 

  2. Faure, P., andHuard, P.,Résultats Nouveaux Relatifs à la Méthode des Centres, Paper Presented at Fourth International Conference on Operations Research, Boston, Massachusetts, 1966.

  3. Bui-Trong-Lieu, andHuard, P.,La Méthode des Centres dans un Espace Topologique, Numerische Mathematik, Vol. 8, pp. 56–67, 1966.

    Google Scholar 

  4. Tremolières, R.,La Méthode des Centres à Troncature Variable, University of Paris, Doctoral Thesis, 1968.

  5. Kleibohm, K.,Äquivalence eines Optimierungsproblems mit Restriktionen und einer Folge von Optimierungsproblemen ohne Restriktionen, Unternehmensforschung, Vol. 11, pp. 111–118, 1967.

    Google Scholar 

  6. Pironneau, O., andPolak, E.,On the Rate of Convergence of Certain Methods of Centers, Mathematical Programming, Vol. 2, pp. 230–257, 1972.

    Google Scholar 

  7. Polak, E.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.

    Google Scholar 

  8. Zangwill, W. I.,Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.

    Google Scholar 

  9. Huard, P.,Programmation Mathématique Convexe, Revue Francaise d'Informatique et de Recherche Opérationnelle, Vol. 7, pp. 43–59, 1968.

    Google Scholar 

  10. Topkis, D. M., andVeinott, A. F.,On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming, SIAM Journal on Control, Vol. 5, pp. 268–279, 1967.

    Google Scholar 

  11. Fiacco, A. V., andMcCormick, G. P.,The Sequential Unconstrained Minimization Technique Without Parameters, Operations Research, Vol. 15, pp. 820–827, 1967.

    Google Scholar 

  12. Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, New York, 1968.

    Google Scholar 

  13. Fiacco, A. V.,Sequential Unconstrained Minimization Methods for Nonlinear Programming, Northwestern University, Evanston, Illinois, PhD Thesis, 1967.

    Google Scholar 

  14. Frisch, K. R.,Principles of Linear Programming with Particular Reference to the Double Gradient Form of the Logarithmic Potential Method, University of Oslo, Institute of Economics, Memorandum, October 18, 1954.

  15. Frisch, K. R.,The Logarithmic Potential Method of Convex Programming, University of Oslo, Institute of Economics, Memorandum, May 13, 1955.

  16. Parisot, G. R.,Résolution Numerique Approchée du Problème de Programmation Linéaire par Application de la Programmation Logarithmique, Revue Francaise d'Informatique et de Recherche Opérationnelle, Vol. 20, pp. 227–259, 1961.

    Google Scholar 

  17. Lootsma, F. A.,Logarithmic Programming: A Method of Solving Nonlinear Programming Problems, Philips Research Reports, Vol. 22, pp. 329–344, 1967.

    Google Scholar 

  18. Lootsma, F. A.,Extrapolation in Logarithmic Programming, Philips Research Reports, Vol. 23, pp. 108–116, 1968.

    Google Scholar 

  19. Carroll, C. W.,The Created Response Surface Technique for Optimizing Nonlinear Restrained Systems, Operations Research, Vol. 9, pp. 169–184, 1961.

    Google Scholar 

  20. Fiacco, A. V., andMcCormick, G. P.,The Sequential Unconstrained Minimization Technique for Nonlinear Programming, A Primal-Dual Method, Management Science, Vol. 10, pp. 360–366, 1964.

    Google Scholar 

  21. Fiacco, A. V., andMcCormick, G. P.,Computational Algorithm for the Sequential Unconstrained Minimization Technique for Nonlinear Programming, Management Science, Vol. 10, pp. 601–617, 1964.

    Google Scholar 

  22. John, F.,Extremum Problems with Inequalities as Subsidiary Conditions, Studies and Essays, Courant Anniversary Volume, John Wiley and Sons (Interscience Publishers), New York, New York, pp. 187–204, 1948.

    Google Scholar 

  23. Mangasarian, O. L.,Pseudo-Convex Functions, SIAM Journal on Control, Vol. 3, pp. 281–290, 1965.

    Google Scholar 

  24. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the Second Berkeley Symposium in Mathematical Statistics and Probability, Edited by J. Neyman, University of California Press, Berkeley, California, 1951.

    Google Scholar 

  25. Mifflin, R.,Convergence Bounds for Nonlinear Programming Algorithms, Mathematical Programming, Vol. 8, pp. 251–271, 1975.

    Google Scholar 

  26. Lootsma, F. A.,Boundary Properties of Penalty Functions for Constrained Minimization, Philips Research Reports, Supplement No. 3, 1970.

  27. Faure, P.,Note Sur la Rapidité de la Convergence de la Méthode des Centres, Electricité de France, Note EDF No. HR-7504/5, 1975.

  28. Lootsma, F. A.,Constrained Optimization via Parameter-Free Penalty Functions, Philips Research Reports, Vol. 23, pp. 424–437, 1968.

    Google Scholar 

  29. Poljak, B. T.,Existence Theorems and Convergence of Minimizing Sequences in Extremum Problems with Restrictions, Soviet Mathematics, Vol. 7, pp. 72–75, 1966.

    Google Scholar 

  30. Topkis, D. M.,Cutting Plane Methods Without Nested Constraint Sets, Operations Research, Vol. 18, pp. 404–413, 1970.

    Google Scholar 

  31. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  32. Lootsma, F. A.,Constrained Optimization via Penalty Functions, Philips Research Reports, Vol. 23, pp. 408–423, 1968.

    Google Scholar 

  33. Kortanek, K. O., andEvans, J. P.,Pseudo-Concave Programming and Lagrange Regularity, Operations Research, Vol. 15, pp. 882–891, 1967.

    Google Scholar 

  34. Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.

    Google Scholar 

  35. Penrose, R.,A Generalized Inverse for Matrices, Proceedings of the Cambridge Philosophical Society, Vol. 51, pp. 406–413, 1955.

    Google Scholar 

  36. Mifflin, R.,Subproblem and Overall Convergence for a Method of Centers Algorithm, Operations Research, Vol. 23, pp. 796–809, 1975.

    Google Scholar 

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Authors and Affiliations

  1. School of Organization and Management, Yale University, New Haven, Connecticut

    R. Mifflin (Assistant Professor)

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  1. R. Mifflin
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Additional information

Communicated by G. L. Nemhauser

This work was supported in part by the National Aeronautics and Space Administration, Predoctoral Traineeship No. NsG(T)-117, and by the National Science Foundation, Grants No. GP-25081 and No. GK-32710.

The author wishes to thank Donald M. Topkis for his valuable criticism of an earlier version of this paper and a referee for his helpful comments.

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Mifflin, R. Rates of convergence for a method of centers algorithm. J Optim Theory Appl 18, 199–228 (1976). https://doi.org/10.1007/BF00935704

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