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On polynomiality of the method of analytic centers for fractional problems

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Abstract

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝn is a closed and bounded convex domain,K ⊂ ℝm is a closed convex cone andA(x):G → ℝn,B(x):G→K are regular enough (say, affine) mappings.

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References

  1. K.M. Anstreicher, “A monotone projective algorithm for fractional linear programming”,Algorithmica 1 (1986) 483–498.

    Article  MATH  MathSciNet  Google Scholar 

  2. C.R. Bector and A. Cambini, “Fractional programming—Some recent results”, in:Generalized Convexity and Fractional Programming with Economic Applications (Pisa, 1988), Lecture Notes in Economics and Mathematical Systems, Vol. 345 (Springer-Verlag, Berlin, 1989) 493–504.

    Google Scholar 

  3. Y. Benadada, J.P. Crouzeix and J. Ferland, “An interval-type algorithm for generalized fractional programming”, in:Generalized Convexity and Fractional Programming with Economic Applications (Pisa, 1988), Lecture Notes in Economics and Mathematical Systems, Vol. 345 (Springer-Verlag, Berlin, 1989) 106–120.

    Google Scholar 

  4. S. Boyd and L. El Ghaoui, “Method of centers for minimizing generalized eigenvalue”,Linear Algebra and Applications (special issue on Numerical Linear Algebra Methods in Control, Signals and Systems) 188 (1993) 63–111.

    Article  MathSciNet  Google Scholar 

  5. S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan,Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994).

    MATH  Google Scholar 

  6. W. Dinkelbach, “On nonlinear fractional programming”,Management Science 13 (1967) 492–498.

    Article  MathSciNet  Google Scholar 

  7. J.A. Ferland and J.Y. Potvin, “Generalized fractional programming: Algorithms and numerical experimentation”,European Journal of Operational Research 20 (1985) 92–101.

    Article  MATH  MathSciNet  Google Scholar 

  8. R.W. Freund and F. Jarre, “An interior-point method for convex fractional programming”, AT&T Numerical Analysis Manuscript No.93-03, Bell Laboratories (Murray Hill, NJ, 1993).

    Google Scholar 

  9. R.W. Freund and F. Jarre, “An interior-point method for multi-fractional programs with convex constraints”, AT&T Numerical Analysis Manuscript No. 93-07, Bell Laboratories (Murray Hill, NJ, 1993).

    Google Scholar 

  10. N. Karmarkar, “A new polynomial-time algorithm for linear programming”,Combinatorica 4 (1984) 373–395.

    Article  MATH  MathSciNet  Google Scholar 

  11. Yu. Nesterov, “Metod lineinogo programmirovania, trebuushii O(n 3 L) operations”,Ekonomika i matem. metody 24 (1988) 25–33 (translated into English as “A method for linear programming which requires O(n 3 L) operations”,Matekon: Translations of Russian and East European Math. Economics).

    Google Scholar 

  12. Yu. Nesterov, “Polinomial'nye metody v lineinom i kvadratichnom programmirovanii”,Izvestija AN SSSR, Tekhnitcheskaya kibernetika 3 (1988).

  13. Yu. Nesterov, “Polinomial'nye iterativnye metody v lineinom i kvadratichnom programmirovanii”, in:Voprosy kibernetiki. Moscow (1988).

  14. Yu. Nesterov and A. Nemirovskii,Interior point polynomial algorithms in Convex Programming: Theory and applications (SIAM, Philadelphia, 1994).

    MATH  Google Scholar 

  15. Yu. Nesterov and A. Nemirovskii, “Technique for constructing self-concordant barriers”, Technical Report. Faculté des Sciences Economiques et Sociales, Department d'Economie Commerciale et Industrielle, Université de Genève, Genève (1993).

    Google Scholar 

  16. Yu. Nesterov and A. Nemirovskii, “An interior point method for generalized linear-fractional problems”,Mathematical Programming 69 (1995) 177–204.

    MathSciNet  Google Scholar 

  17. J. Renegar, “A polynomial time algorithm, based on Newton's method, for linear programming”,Mathematical Programming 40 (1988) 59–93.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Schaible, “Bibliography in fractional programming”,Zeitschrift für Operations Research 27 (1983) 211–241.

    Article  Google Scholar 

  19. S. Schaible, “Multi-ratio fractional programming—A survey”, in:Optimization, Parallel Processing and Applications, Lecture Notes in Economics and Mathematical Systems, Vol. 304 (Springer-Verlag, Berlin, 1988) 57–66.

    Google Scholar 

  20. G. Sonnevend, “An ‘analytical centre’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming”, in:System Modelling and Optimization (Budapest, 1985), Lecture Notes in Control and Information Sciences, Vol. 84 (Springer Verlag, Berlin, 1985) 866–875.

    Chapter  Google Scholar 

  21. J. von Neumann, “A model of general economic equilibrium”,Review of Economic Studies 13 (1945) 1–9.

    Article  Google Scholar 

  22. Y. Ye, “On the von Neumann economic growth problem”, Working Paper Series No. 92-9, Department of Management Sciences, The University of Iowa, Iowa City, IA 52242 (1992).

    Google Scholar 

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Authors and Affiliations

  1. Central Economic & Mathematical Institute of the Russian Academy of Sciences, 32 Krasikova Str., 117418, Moscow, Russia

    A. Nemirovskii

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  1. A. Nemirovskii
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This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences

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Nemirovskii, A. On polynomiality of the method of analytic centers for fractional problems. Mathematical Programming 73, 175–198 (1996). https://doi.org/10.1007/BF02592102

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  • DOI: https://doi.org/10.1007/BF02592102

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