Certain algorithms that are known to converge can be renormalized or "blown up" at each iteration so that their local behavior can be seen. This creates dynamical systems that we can study with modern tools, such as ergodic theory, chaos, special attractors, and Lyapounov exponents. Furthermore, we can translate the rates of convergence into less studied exponents known as Renyi entropies.
This all feeds back to suggest new algorithms with faster rates of convergence. For example, in line-search, we can improve upon the Golden Section algorithm with new classes of algorithms that have their own special-and sometimes chaotic-dynamical systems. The ellipsoidal algorithms of linear and convex programming have fast, "deep cut" versions whose dynamical systems contain cyclic attractors. And ordinary steepest descent has, buried within, a beautiful fractal that controls the gateway to a special two-point attractor. Faster "relaxed" versions exhibit classical period doubling.
Dynamical Search presents a stimulating introduction to a brand new field - the union of dynamical systems and optimization. It will prove fascinating and open doors to new areas of investigation for researchers in both fields, plus those in statistics and computer science.
Table of Contents
Consistency in Discrete Search
Consistency in Linear and Convex Programming
Towards a Dynamical System Representation
Renormalisation in Line-Search Algorithms
Renormalisation of Ellipsoid Algorithms
Renormalisation in the Steepest Descent Algorithm
RATES OF CONVERGENCE
Ergodic Rates of Convergence
Characteristics of Average Performances
Characteristics of Worst-Case Performances
One-Dimensional Piecewise Linear Mappings
First-Order Line Search
Continued Fraction Expansion and the Gauss Map
Ergodically Optimal Second-Order Line Search Algorithm
Midpoint and Window Algorithms
Algorithms Based on Section Invariant Numbers
Comparisons of GS, GS4, GS40 and Window Algorithms
Volume-Optimal Outer and Inner Ellipsoids
Ergodic Behaviour of the Outer-Ellisoid Algorithm
STEEPEST DESCENT ALGORITHM
Attraction to a Two-Dimensional Plane
Stability of Attractors
Rate of Convergence
Steepest Descent with Relaxation
"…a fascinating book which links optimization algorithms with the properties of certain dynamical systems. This link allows one to better understand the optimization algorithms, and to ultimately construct more efficient versions of them…"
-Short Book Reviews of the ISI
"There is a great need for a text which provides this deeper level of consideration and rigour."
--Stephen Clark, University of Leeds, UK
"…an established research programme which aims to link the areas of dynamical systems and search theory. … The book is of value to those researchers and academics who have an interest in this area and wish to explore the research ideas that it contains in a more extensive coverage than is usually found in reports and journal papers."
-The Statistician, Vol. 50, Part 1, 2001