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An implementation of the method of analytic centers

  • Algorithmes For Optimization And Optimal Control
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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 111))

Abstract

In this report we present an implementation of a new method for the solution of convex programming problems based on the use of a homotopy path leading to an optimal solution through analytic centers. With a number of numerical test results on linear programming problems we indicate that this method has a favorable convergence behaviour and a substantial acceleration can be achieved by using a suitable (higher than first or second order) extrapolation for following the homotopy path.

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References

  1. I. Adler, M.G.C. Resende, G. Veiga: An Implementation of Karmarkar's Algorithm for Linear Programming, ORC Report May 1986, Dept. of Ind. Engin. and OR, University of California, Berkeley, CA 94720

    Google Scholar 

  2. D. A. Bayer and J. C. Lagarias: Karmarkar's linear programming algorithm and Newton's method, June 1987, preprint

    Google Scholar 

  3. R. Bulirsch, J. Stoer: Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math. 8, 1–13 (1966)

    Google Scholar 

  4. P. Huard and B.T. Liêu: La Méthode des Centres dans un Espace Topologique, Numer. Math. 8 (1966), 56–67

    Google Scholar 

  5. C.C. Gonzaga: An Algorithm for Solving Linear Programming Problems in o(n 3 L) Operations, Memo No. UCB/ERL M87/10 5 March 1987, College of Engineering, University of California, Berkeley, CA 94720

    Google Scholar 

  6. H. Imai: Extensions of the Multiplicative Penalty Function Method for Linear Programming, March 1986, to appear in Journal of the Oper. Res. Society of Japan.

    Google Scholar 

  7. M. Iri and H. Imai: A Multiplicative Barrier Function Method for Linear Programming, Aug 1985 — July 1986, Algorithmica (1986) 1: 455–482

    Google Scholar 

  8. F. Jarre: On the Convergence of the Method of Analytic Centers when applied to Convex Quadratic Programs, Report No. 35, Dec. 1987, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft — Anwendungsbezogene Optimierung und Steuerung, revised Feb. 1988, submitted to Math. Prog.

    Google Scholar 

  9. N. Karmarkar: A New Polynomial-Time Algorithm for Linear Programming, Combinatorica, 4 (1984), 373–395

    Google Scholar 

  10. V. Klee and G.J. Minty: How good is the simplex algorithm?, 1972 Inequalities III (O. Sisha, ed.) Academic Press, New York

    Google Scholar 

  11. M. Kojima, S. Mizuno, A. Yoshise: A Polynomial-Time Algorithm for a Class of Linear Complementarity Problems, June 1987, preprint, Dept. of Information Sciences, Tokyo Institute of Technology

    Google Scholar 

  12. J. Renegar: A Polynomial Time Algorithm Based on Newton's Method for Linear Programming, July 1986, Dept. of Mathematics, Colorado State University.

    Google Scholar 

  13. G. Sonnevend: An Analytic Centre for Polyhedrons and New Classes of Global Algorithms for Linear (Smooth, Convex) Programming, Proc. 12th IFIP Conf. on System Modelling and Optimization, Lecture Notes in Control and Information Sciences, vol. 84.

    Google Scholar 

  14. G. Sonnevend: A New Method for Solving a Set of Linear (Convex) Inequalities and its Applications, in Proc. of the 5th IFAC-IFORS Conf. Budapest, June 1986, ed by B. Martos, Pergamon Press Oxford, 1987

    Google Scholar 

  15. G. Sonnevend: New algorithms in convex programming based on a notion of „centre“ (for solving systems of analytic inequalities) and on rational extrapolation. Proc. of the 4th French-German Conf. on Optimization, Irsee, April 1986, ed. by K-H.Hoffmann et al. Int. Series of Numerical Math. vol 84. Birkhäuser Verlag (1987)

    Google Scholar 

  16. G. Sonnevend and J. Stoer: Global Ellipsoidal Approximations and Homotopy Methods for Solving Convex Analytic Programs, Report No. 40, Jan. 1988, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft — Anwendungsbezogene Optimierung und Steuerung, Inst. f. Ang. Mathematik und Statistik, Universität Würzburg

    Google Scholar 

  17. K. Tanabe, Complementarity — enforcing centered Newton method for mathematical programming, preprint, The Institute of Statistical Mathematics, 4 — 6 — 7, Minamiazabu, Minatoku, Tokyo, Japan 106

    Google Scholar 

  18. P. Vaidya, An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations, Report from AT&T Bell Laboratories, Murray Hill, NJ, (1987)

    Google Scholar 

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A. Bensoussan J. L. Lions

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© 1988 Springer-Verlag

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Jarre, F., Sonnevend, G., Stoer, J. (1988). An implementation of the method of analytic centers. In: Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences, vol 111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0042223

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  • DOI: https://doi.org/10.1007/BFb0042223

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-19237-4

  • Online ISBN: 978-3-540-39161-6

  • eBook Packages: Springer Book Archive

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