Numerical Representations of a Universal Subspace Flow for Linear Programs

In 1991, Sonnevend, Stoer, and Zhao [Math. Programming 52 (1991) 527--553] have shown that the central paths of strictly feasible instances of linear programs generate curves on the Grassmannian that satisfy a universal ordinary differential equation. Instead of viewing the Grassmannian $Grassmn$ as the set of all $n imes n$ projection matrices of rank $m$, we view it as the set $STmn$ of all full column rank $n imes m$ matrices, quotiented by the right action of the general linear group $GL(m)$. We propose a class of flows in $STmn$ that project to the flow on the Grassmannian. This approach requires much less storage space when $ngg m$ (i.e., there are many more constraints than variables in the dual formulation). One of the flows in $STmn$, that leaves invariant the set of orthonormal matrices, turns out to be a particular version of a matrix differential equation known as Oja's flow. We also point out that the flow in the set of projection matrices admits a double bracket expression.

[1]  Super Brockett Equations:¶A Graded Gradient Integrable System , 2001 .

[2]  Tin-Yau Tam.  Gradient flows and double bracket equations , 2004 .

[3]  John B. Moore,et al.  Global analysis of Oja's flow for neural networks , 1994, IEEE Trans. Neural Networks.

[4]  Josef Stoer,et al.  On the complexity of following the central path of linear programs by linear extrapolation II , 1991, Math. Program..

[5]  Arieh Iserles.  On the Discretization of Double-Bracket Flows , 2002, Found. Comput. Math..

[6]  Anthony M. Bloch,et al.  On the optimality of double-bracket flows , 2004, Int. J. Math. Math. Sci..

[7]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[8]  Kenneth R. Driessel,et al.  Constructing symmetric nonnegative matrices with prescribed eigenvalues by differential equations , 1991 .

[9]  B. M. Fulk.  MATH , 1992 .

[10]  P. Absil,et al.  Continuous-time subspace flows related to the symmetric eigenproblem ∗ , 2008 .

[11]  R. Brockett.  Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[12]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[13]  Moody T. Chu,et al.  Constructing a Hermitian Matrix from Its Diagonal Entries and Eigenvalues , 1995, SIAM J. Matrix Anal. Appl..

[14]  Erkki Oja,et al.  Neural Networks, Principal Components, and Subspaces , 1989, Int. J. Neural Syst..

[15]  P. Absil,et al.  Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation , 2004 .

[16]  Antonella Zanna,et al.  Numerical solution of isospectral flows , 1997, Math. Comput..

[17]  Gongyun Zhao,et al.  Representing the space of linear programs as the Grassmann manifold , 2009, Math. Program..

[18]  E. Oja.  Simplified neuron model as a principal component analyzer , 1982, Journal of mathematical biology.

[19]  Sabine Van Huffel,et al.  A Geometric Newton Method for Oja's Vector Field , 2008, Neural Computation.

[20]  Iven M. Y. Mareels,et al.  A dual purpose principal and minor component flow , 2005, Syst. Control. Lett..

[21]  Fernando Casas,et al.  Numerical integration methods for the double-bracket flow , 2004 .

[22]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[23]  R. Brockett,et al.  Completely integrable gradient flows , 1992 .

[24]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.