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Linear Programs and Convex Hulls Over Fields of Puiseux Fractions

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Mathematical Aspects of Computer and Information Sciences (MACIS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9582))

Abstract

We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry.

M. Joswig—Partially supported by Einstein Foundation Berlin and Deutsche Forschungsgemeinschaft (DFG) within the Priority Program 1489 “Experimental Methods in Algebra, Geometry and Number Theory”.

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References

  1. Allamigeon, X., Benchimol, P., Gaubert, S., Joswig, M.: Long and winding central paths, preprint (2014). arXiv:1405.4161

  2. Allamigeon, X., Benchimol, P., Gaubert, S., Joswig, M.: Tropicalizing the simplex algorithm. SIAM J. Discrete Math. 29(2), 751–795 (2015). http://dx.doi.org/10.1137/130936464

    Article  MathSciNet  MATH  Google Scholar 

  3. Allamigeon, X., Gaubert, S., Goubault, É.: The tropical double description method. In: STACS 2010: 27th International Symposium on Theoretical Aspects of Computer Science, LIPIcs. Leibniz International Proceedings in Informatics, vol. 5, pp. 47–58. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2010)

    Google Scholar 

  4. Allamigeon, X., Gaubert, S., Goubault, É.: Computing the vertices of tropical polyhedra using directed hypergraphs. Discrete Comput. Geom. 49(2), 247–279 (2013). http://dx.doi.org/10.1007/s00454-012-9469-6

    Article  MathSciNet  MATH  Google Scholar 

  5. Benchimol, P.: Tropical aspects of linear programming. Theses, École Polytechnique, December 2014. https://hal-polytechnique.archives-ouvertes.fr/tel-01198482

  6. Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997). http://dx.doi.org/10.1006/jsco.1996.0125

    Article  MathSciNet  MATH  Google Scholar 

  7. Charnes, A., Kortanek, K.O.: On classes of convex and preemptive nuclei for \(n\)-person games. In: Proceedings of the Princeton Symposium on Mathematical Programming (Princeton University, 1967). pp. 377–390. Princeton University Press, Princeton (1970)

    Google Scholar 

  8. Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton, N.J. (1963)

    MATH  Google Scholar 

  9. Develin, M., Yu, J.: Tropical polytopes and cellular resolutions. Experiment. Math. 16(3), 277–291 (2007). http://projecteuclid.org/euclid.em/1204928529

    Article  MathSciNet  MATH  Google Scholar 

  10. Deza, A., Terlaky, T., Zinchenko, Y.: Central path curvature and iteration-complexity for redundant Klee-Minty cubes. In: Gao, D.Y., Sherali, H.D. (eds.) Advances in Applied Mathematics and Global Optimization. Adv. Mech. Math., vol. 17, pp. 223–256. Springer, New York (2009). http://dx.doi.org/10.1007/978-0-387-75714-8_7

    Chapter  Google Scholar 

  11. Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer-Verlag, Berlin (1987). http://dx.doi.org/10.1007/978-3-642-61568-9

    Book  MATH  Google Scholar 

  12. von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  13. Gaubert, S., Katz, R.D.: Minimal half-spaces and external representation of tropical polyhedra. J. Algebraic Combin. 33(3), 325–348 (2011). http://dx.doi.org/10.1007/s10801-010-0246-4

    Article  MathSciNet  MATH  Google Scholar 

  14. Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes–combinatorics and computation (Oberwolfach, 1997). DMV Sem., pp. 43–73. Birkhäuser, Basel (2000)

    Google Scholar 

  15. Goldfarb, D., Sit, W.Y.: Worst case behavior of the steepest edge simplex method. Discrete Appl. Math. 1(4), 277–285 (1979). http://dx.doi.org/10.1016/0166-218X(79)90004-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Jeroslow, R.G.: Asymptotic linear programming. Oper. Res. 21(5), 1128–1141 (1973). http://dx.doi.org/10.1287/opre.21.5.1128

    Article  MathSciNet  MATH  Google Scholar 

  17. Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: On polyhedral projection and parametric programming. J. Optim. Theor. Appl. 138(2), 207–220 (2008). http://dx.doi.org/10.1007/s10957-008-9384-4

    Article  MathSciNet  MATH  Google Scholar 

  18. Joswig, M.: Beneath-and-beyond revisited. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 1–21. Springer, Berlin (2003)

    Chapter  Google Scholar 

  19. Joswig, M.: Tropical convex hull computations. In: Litvinov, G.L., Sergeev, S.N. (eds.) Tropical and Idempotent Mathematics, Contemporary Mathematics, vol. 495, pp. 193–212. American Mathematical Society, Providence (2009)

    Chapter  Google Scholar 

  20. Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Inequalities, III (Proceedings of Third Symposium, University of California, Los Angeles, California, 1969; dedicated to the memory of Theodore S. Motzkin), pp. 159–175. Academic Press, New York (1972)

    Google Scholar 

  21. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence, RI (2015)

    MATH  Google Scholar 

  22. Mannaa, B., Coquand, T.: Dynamic Newton-Puiseux theorem. J. Log. Anal. 5, 22 (2013). Paper 5

    MathSciNet  MATH  Google Scholar 

  23. Markwig, T.: A field of generalised Puiseux series for tropical geometry. Rend. Semin. Mat. Univ. Politec. Torino 68(1), 79–92 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Maslov, V.P.: On a new superposition principle for optimization problem. In: Séminaire sur les équations aux dérivées partielles, pp. 1985–1986, Exp. No. XXIV, 14. École Polytech., Palaiseau (1986)

    Google Scholar 

  25. Opfer, T.: Entwicklung eines exakten rationalen dualen Simplex-Lösers. Master’s thesis, TU Darmstadt (2011)

    Google Scholar 

  26. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, vol. 26. The University Press, Clarendon Press, Oxford (2002)

    Google Scholar 

  27. Salzmann, H., Grundhöfer, T., Hähl, H., Löwen, R.: The classical fields, Encyclopedia of Mathematics and its Applications, vol. 112. Cambridge University Press, Cambridge (2007). http://dx.doi.org/10.1017/CBO9780511721502

    MATH  Google Scholar 

  28. Tarski, A.: A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica, California (1948)

    MATH  Google Scholar 

  29. The MathWorks Inc.: MATLAB, version 8.4.0.150421 (R2014b). Natick, Massachusetts (2014)

    Google Scholar 

  30. van der Waerden, B.L.: Algebra II. Unter Benutzung von Vorlesungen von E. Artin und E. Noether, 6th edn. Springer, Berlin (1993). Mit einem Geleitwort von Jürgen Neukirch

    MATH  Google Scholar 

  31. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag, New York (1995)

    MATH  Google Scholar 

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Acknowledgments

We thank Thomas Opfer for contributing to and maintaining within the polymake project his implementation of the dual simplex method, originally written for his Master’s Thesis [25].

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Authors and Affiliations

  1. Institut für Mathematik, TU Berlin, MA 6-2, Str. des 17. Juni 136, 10623, Berlin, Germany

    Michael Joswig, Georg Loho, Benjamin Lorenz & Benjamin Schröter

Authors
  1. Michael Joswig
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  2. Georg Loho
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  3. Benjamin Lorenz
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  4. Benjamin Schröter
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Corresponding authors

Correspondence to Michael Joswig , Georg Loho , Benjamin Lorenz or Benjamin Schröter .

Editor information

Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

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Joswig, M., Loho, G., Lorenz, B., Schröter, B. (2016). Linear Programs and Convex Hulls Over Fields of Puiseux Fractions. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_37

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  • DOI: https://doi.org/10.1007/978-3-319-32859-1_37

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