Dynamical Search: Applications of Dynamical Systems in Search and Optimization

Luc Pronzato, Henry P. Wynn, Anatoly A Zhigljavsky

August 27, 1999 by Chapman and Hall/CRC
Reference - 240 Pages
ISBN 9780849303364 - CAT# C0336
Series: Chapman & Hall/CRC Interdisciplinary Statistics


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  • Provides quick access to the state-of-the-art of dynamical systems and optimisation
  • Contains many figures that provide new insight into dynamical behavior
  • Offers results that can serve as a starting point for new research
  • Includes a rich supply of examples plus appendices covering entropies, ergodic theory, and section-invariant numbers
  • Stands as a unique work-the first that addresses dynamical systems and optimization
  • Summary

    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.


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