The Central Curve in Linear Programming

Foundations of Computational Mathematics

Volumn 12, Issue 4, 2012, Pages 509-540

  de Loera, Jesús A ^a   Sturmfels, Bernd ^b   Vinzant, Cynthia ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Central path; Complementary slackness; Curvature; Degree; Gauss map; Gr bner basis; Hyperbolic polynomial; Hyperplane arrangement; Interior point methods; Linear programming; Matroid; Prime ideal; Projective variety; Total curvature; Tutte polynomial

Indexed keywords

EID: 84864304119     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-012-9127-7     Document Type: Article

References (36)

1. I. Adler, The expected number of pivots needed to solve parametric linear programs and the efficiency of the self-dual simplex method, Technical Report, University of California at Berkeley, CA (1983).
2. A. Bachem, W. Kern, Linear Programming Duality. An Introduction to Oriented Matroids, Universitext (Springer, Berlin, 1992).
3. D. Bayer, J. Lagarias, The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories, Trans. Am. Math. Soc. 314, 499-526 (1989).
4. D. Bayer, J. Lagarias, The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories, Trans. Am. Math. Soc. 314, 527-581 (1989).
5. R. Benedetti, M. Dedò, A geometric inequality for the total curvature of plane curves, Geom. Dedic. 22, 105-115 (1987).
6. A. Berget, Products of linear forms and Tutte polynomials, Eur. J. Comb. 31, 1924-1935 (2010).
7. A. Björner, The homology and shellability of matroids and geometric lattices, in Matroid Applications. Encyclopedia Math. Appl., vol. 40 (Cambridge Univ. Press, Cambridge, 1992), pp. 226-283.
8. K. H. Borgwardt, Probabilistic analysis of the simplex method, in Mathematical Developments Arising from Linear Programming, ed. by J. C. Lagarias, M. J. Todd. Contemporary Mathematics, vol. 114 (Am. Math. Soc., Providence, 1990), pp. 21-34.
9. E. Brugallé, L. López de Medrano, Inflection points of real and tropical plane curves, J. Singularities (2012). doi: 10. 5427/jsing. 2012. 4e.
10. T. Brylawski, J. Oxley, The Tutte polynomial and its applications, in Matroid Applications. Encyclopedia Math. Appl., vol. 40 (Cambridge Univ. Press, Cambridge, 1992), pp. 123-225.
11. J. P. Dedieu, G. Malajovich, M. Shub, On the curvature of the central path for linear programming theory, Found. Comput. Math. 5, 145-171 (2005).
12. A. Deza, T. Terlaky, Y. Zinchenko, A continuous d-step conjecture for polytopes, Discrete Comput. Geom. 41, 318-327 (2009).
13. A. Deza, T. Terlaky, Y. Zinchenko, Polytopes and arrangements: Diameter and curvature, Oper. Res. Lett. 36(2), 215-222 (2008).
14. I. V. Dolgachev, Classical Algebraic Geometry: A Modern View (Cambridge University Press, Cambridge, 2012).
15. D. Grayson, M. Stillman, Macaulay 2, a software system for research in algebraic geometry, available at www. math. uiuc. edu/Macaulay2/.
16. P. Griffiths, On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41, 775-814 (1974).
17. M. Haimovich, The simplex algorithm is very good!: On the expected number of pivot steps and related properties of random linear programs. Technical Report, Columbia University, New York (1983).
18. F. Klein, Eine neue Relation zwischen den Singularitäten einer algebraischer Curve, Math. Ann. 10, 199-210 (1876).
19. E. L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).
20. A. Lewis, P. Parrilo, M. Ramana, The Lax conjecture is true, Proc. Am. Math. Soc. 133, 2495-2499 (2005).
21. N. Megiddo, M. Shub, Boundary behavior of interior point algorithms in linear programming. Math. Oper. Res., 14(1), 97-146 (1989).
22. R. Monteiro, T. Tsuchiya, A strong bound on the integral of the central path curvature and its relationship with the iteration complexity of primal-dual path-following LP algorithms, Math. Program., Ser. A115, 105-149 (2008).
23. R. Piene, Numerical characters of a curve in projective space, in Real and Complex Singularities. Proc. Ninth Nordic Summer School Sympos. Math., Oslo, 1976 (1977), pp. 475-495. Sijthoff and Noordhoff.
24. N. Proudfoot, D. Speyer, A broken circuit ring, Beiträge Algebra Geom. 47, 161-166 (2006).
25. J. Renegar, Hyperbolic programs, and their derivative relaxations, Found. Comput. Math. 6, 59-79 (2006).
26. C. Roos, T. Terlaky, J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior Point Approach 2nd edn. (Springer, New York, 2006).
27. P. Rostalski, B. Sturmfels, Dualities in convex algebraic geometry. Rend. Mat. Appl. (7)30(3-4), 285-327 (2010).
28. G. Sonnevend, J. Stoer, G. Zhao, On the complexity of following the central path of linear programs by linear extrapolation. II. Math. Program., Ser. B52, 527-553 (1991).
29. R. P. Stanley, Combinatorics and Commutative Algebra, 2nd edn. Progress in Mathematics, vol. 41 (Birkhäuser, Boston, 1996).
30. B. Sturmfels, C. Uhler, Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry, Ann. Inst. Stat. Math. 62, 603-638 (2010).
31. H. Terao, Algebras generated by reciprocals of linear forms, J. Algebra250, 549-558 (2002).
32. R. J. Vanderbei, Linear programming: foundations and extensions, in Operations Research & Management Science, 3rd edn. International Series, vol. 114 (Springer, New York, 2008).
33. S. Vavasis, Y. Ye, A primal-dual interior-point method whose running time depends only on the constraint matrix, Math. Program., Ser. A74, 79-120 (1996).
34. V. Vinnikov, Self-adjoint determinantal representations of real plane curves, Math. Ann. 296(3), 453-479 (1993).
35. T. Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Am. Math. Soc., 1, 154 (1975).
36. G. Zhao, J. Stoer, Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals, Appl. Math. Optim. 27, 85-103 (1993).