Abstract
A Lax pair representation of a nonlinear dynamical system is presented which emerges in linear programming as a continuous version of Karmarkar’s projective scaling algorithm. Karmarkar’s continuous trajectory is also shown to be a gradient system. An expression of solution in a rational form is found by integrating the associated dynamical system. Fixed points of the trajectory also studied.
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V.I. Arnol’d and A.B. Givental’, Symplectic geometry. Dynamical Systems IV (eds. V.I. Arnol’d and S.P. Novikov), Encyc. of Math. Sci. Vol. 4, Springer-Verlag, Berlin, 1990, 1–136.
D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories. Trans. Amer. Math. Soc.,314 (1989), 499–526.
D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories. Trans. Amer. Math. Soc.,314 (1989), 527–581.
A.M. Bloch, Steepest descent, linear programming, and Hamiltonian flows. Mathematical Developments Arising from Linear Programming (eds. J.C. Lagarias and M.J. Todd), Contemp. Math., Vol. 114, Amer. Math. Soc., Providence, 1990, 77–88.
R.W. Brockett, Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. Linear Algebra Appl.,146 (1991), 79–91; also Proc. 27th IEEE Conf. on Decision and Control, 1988, 799–803.
I.I. Dikin, Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl.,8 (1967), 674–675.
L. Faybusovich, Hamiltonian structure of dynamical systems which solve linear programming problems. Physica, D53 (1991), 217–232.
M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Pure and Appl. Math., Vol. 5, Academic Press, New York, 1974.
M. Iri, Integrability of vector and multivector fields associated with interior point methods for linear programming. Math. Programming,52 (1991), 511–525.
N. Karmarkar, A new polynomial time algorithm for linear programming. Combinatorica,4 (1984), 373–395.
N. Karmarkar, Riemannian geometry underlying interior-point methods for linear programming. Mathematical Developments Arising from Linear Programming (eds. J.C. Lagarias and M.J. Todd), Contemp. Math., Vol. 114, Amer. Math. Soc., Providence, 1990, 51–75.
J.C. Lagarias, The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry. Trans. Amer. Math. Soc.,320 (1990), 193–225.
N. Megiddo and M. Shub, Boundary behavior of interior point algorithms in linear programming. Math. Oper. Res.,14 (1989), 97–146.
J. Moser, Finitely many points on the line under the influence of an exponential potential — An integrable system. Dynamical Systems, Theory and Applications (ed. J. Moser), Lecture Notes in Phys., Vol. 38, Springer-Verlag, Berlin, 1975, 467–497.
Y. Nakamura, Lax equations associated with a least squares problem and compact Lie algebras. Progress in Differential Geometry (ed. K. Shiohama), Adv. Stud. Pure Math., Vol. 22, Math. Soc. Japan, Tokyo, 1993, 213–229.
Y. Nakamura, A new nonlinear dynamical system that leads to eigenvalues. Japan J. Indust. Appl. Math.,9 (1992), 133–139.
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Nakamura, Y. Lax pair and fixed point analysis of Karmarkar’s projective scaling trajectory for linear programming. Japan J. Indust. Appl. Math. 11, 1–9 (1994). https://doi.org/10.1007/BF03167209
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DOI: https://doi.org/10.1007/BF03167209