**Fields Institute Communications**

Volume: 3;
1994;
155 pp;
Hardcover

MSC: Primary 34; 70; 93; 65; 90;

Print ISBN: 978-0-8218-0255-7

Product Code: FIC/3

List Price: $92.00

AMS Member Price: $73.60

MAA Member Price: $82.80

**Electronic ISBN: 978-1-4704-2971-3
Product Code: FIC/3.E**

List Price: $92.00

AMS Member Price: $73.60

MAA Member Price: $82.80

# Hamiltonian and Gradient Flows, Algorithms and Control

Share this page *Edited by *
*Anthony Bloch*

A co-publication of the AMS and Fields Institute

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

## Hamiltonian and Gradient Flows, Algorithms and Control

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Resonant geometric phases for soliton equations 110
- Schur flows for orthogonal Hessenberg matrices 2736
- Sub-Riemannian optimal control problems 3544
- Systems of hydrodynamic type, connected with the toda lattice and the Volterra model 4958
- The double bracket equation as the solution of a variational problem 6978
- Integration and visualization of matrix orbits on the connection machine 7786
- A list of matrix flows with applications 8796
- The Gibbs variational principle, gradient flows, and interior-point methods 99108
- Optimization techniques on Riemannian manifolds 113122
- On the number of real roots of a sparse polynomial system 137146
- Gradient flows for local minima of combinatorial optimization problems 145154
- Back Cover Back Cover1166