This volume brings together ideas from several areas of mathematics that have traditionally been rather disparate. The conference at The Fields Institute which gave rise to these proceedings was intended to encourage such connections. One of the key interactions occurs between dynamical systems and algorithms, one example being the by now classic observation that the QR algorithm for diagonalizing matrices may be viewed as the time-1 map of the Toda lattice flow. Another link occurs with interior point methods for linear programming, where certain smooth flows associated with such programming problems have proved valuable in the analysis of the corresponding discrete problems. More recently, other smooth flows have been introduced which carry out discrete computations (such as sorting sets of numbers) and which solve certain least squares problems. Another interesting facet of the flows described here is that they often have a dual Hamiltonian and gradient structure, both of which turn out to be useful in analyzing and designing algorithms for solving optimization problems. This volume explores many of these interactions, as well as related work in optimal control and partial differential equations.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Mathematicians and engineers interested in dynamics, optimization, control theory, Hamiltonian and integrable systems, numerical analysis, and linear programming.

Table of Contents

• M. S. Alber and J. E. Marsden -- Resonant geometric phases for soliton equations
• G. S. Ammar and W. B. Gragg -- Schur flows for orthogonal Hessenberg matrices
• A. M. Bloch, P. E. Crouch, and T. S. Ratiu -- Sub-Riemannian optimal control problems
• O. I. Bogoyavlenskii -- Systems of hydrodynamic type, connected with the toda lattice and the Volterra model
• R. W. Brockett -- The double bracket equation as the solution of a variational problem
• J.-P. Brunet -- Integration and visualization of matrix orbits on the connection machine
• M. T.-C. Chu -- A list of matrix flows with applications
• L. E. Faybusovich -- The Gibbs variational principle, gradient flows, and interior-point methods
• S. T. Smith -- Optimization techniques on Riemannian manifolds
• B. Sturmfels -- On the number of real roots of a sparse polynomial system
• W. S. Wong -- Gradient flows for local minima of combinatorial optimization problems