Dynamical Search - Applications of Dynamical Systems in Search and Optimization: Interdisciplinary Statistics

INTRODUCTION CONSISTENCY Consistency in Discrete Search Line-Search Algorithms Consistency in Linear and Convex Programming RENORMALISATION Towards a Dynamical System Representation Renormalisation in Line-Search Algorithms Renormalisation of Ellipsoid Algorithms Renormalisation in the Steepest Descent Algorithm Square Algorithm RATES OF CONVERGENCE Ergodic Rates of Convergence Characteristics of Average Performances Characteristics of Worst-Case Performances Counting Characteristic One-Dimensional Piecewise Linear Mappings LINE-SEARCH ALGORITHMS First-Order Line Search Continued Fraction Expansion and the Gauss Map Golden-Section Algorithm Ergodically Optimal Second-Order Line Search Algorithm Symmetric Algorithms Midpoint and Window Algorithms Algorithms Based on Section Invariant Numbers Comparisons of GS, GS4, GS40 and Window Algorithms ELLIPSOID 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 Appendix Entropies Ergodic Theory Section-Invariant Numbers References Author Index Subject Index