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Abstract Convexity and Global Optimization

  • Book
  • © 2000

Overview

Authors:
  1. Alexander Rubinov

    1. School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia

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Part of the book series: Nonconvex Optimization and Its Applications (NOIA, volume 44)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xviii
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  2. An Introduction to Abstract Convexity

    • Alexander Rubinov
    Pages 1-14

  3. Elements of Monotonic Analysis: IPH Functions and Normal Sets

    • Alexander Rubinov
    Pages 15-73

  4. Elements of Monotonic Analysis: Monotonic Functions

    • Alexander Rubinov
    Pages 75-112

  5. Application to Global Optimization: Lagrange and Penalty Functions

    • Alexander Rubinov
    Pages 113-151

  6. Elements of Star-Shaped Analysis

    • Alexander Rubinov
    Pages 153-227

  7. Supremal Generators and Their Applications

    • Alexander Rubinov
    Pages 229-270

  8. Further Abstract Convexity

    • Alexander Rubinov
    Pages 271-343

  9. Application to Global Optimization: Duality

    • Alexander Rubinov
    Pages 345-397

  10. Application to Global Optimization: Numerical Methods

    • Alexander Rubinov
    Pages 399-470

  11. Back Matter

    Pages 471-493
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Keywords

About this book

Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac­ complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu­ ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh­ bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func­ tion. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.

Reviews

'This book, written by one of the leading contributors in the field, is an up-to-date and very valuable reference. It will be precious to any researcher working in the field on theoretical aspects and applications as well. It opens new ways in global optimization.'
Mathematical Reviews (2002i)
'The book will be very useful for experts in optimisation and aspects of functional and numerical analysis and related topics. This is an excellent introduction to those who are interested in the study of abstract convexity and its applications and to very interesting and highly applicable generalisations on convexity.'
Australian Mathematical Society GAZETTE, August (2002)

Authors and Affiliations

  • School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia

    Alexander Rubinov

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