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Book DetailsFields Institute CommunicationsVolume: 3; 1994; 155 ppMSC: Primary 34; 70; 93; 65; 90;
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 time1 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 copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
ReadershipMathematicians and engineers interested in dynamics, optimization, control theory, Hamiltonian and integrable systems, numerical analysis, and linear programming.

Table of Contents

Chapters

Mark Alber and Jerrold Marsden — Resonant geometric phases for soliton equations

Gregory Ammar and William Gragg — Schur flows for orthogonal Hessenberg matrices

Anthony Bloch, Peter Crouch and Tudor Ratiu — SubRiemannian optimal control problems

O. Bogoyavlenskii — Systems of hydrodynamic type, connected with the toda lattice and the Volterra model

Roger Brockett — The double bracket equation as the solution of a variational problem

JeanPhilippe Brunet — Integration and visualization of matrix orbits on the connection machine

Moody Chu — A list of matrix flows with applications

Leonid Faybusovich — The Gibbs variational principle, gradient flows, and interiorpoint methods

Steven Smith — Optimization techniques on Riemannian manifolds

Bernd Sturmfels — On the number of real roots of a sparse polynomial system

Wing Wong — Gradient flows for local minima of combinatorial optimization problems


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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 time1 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 copublished with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Mathematicians and engineers interested in dynamics, optimization, control theory, Hamiltonian and integrable systems, numerical analysis, and linear programming.

Chapters

Mark Alber and Jerrold Marsden — Resonant geometric phases for soliton equations

Gregory Ammar and William Gragg — Schur flows for orthogonal Hessenberg matrices

Anthony Bloch, Peter Crouch and Tudor Ratiu — SubRiemannian optimal control problems

O. Bogoyavlenskii — Systems of hydrodynamic type, connected with the toda lattice and the Volterra model

Roger Brockett — The double bracket equation as the solution of a variational problem

JeanPhilippe Brunet — Integration and visualization of matrix orbits on the connection machine

Moody Chu — A list of matrix flows with applications

Leonid Faybusovich — The Gibbs variational principle, gradient flows, and interiorpoint methods

Steven Smith — Optimization techniques on Riemannian manifolds

Bernd Sturmfels — On the number of real roots of a sparse polynomial system

Wing Wong — Gradient flows for local minima of combinatorial optimization problems