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Singularities in Optimization Problems: the Maximum Function

  • Vladimir Igorevich Arnold

Abstract

Many singularities, bifurcations, and catastrophes (Jumps) arise in all problems in which extrema (maxima and minima) are sought, problems in optimization, control theory and decision theory. For instance, suppose we have to find x such that the value of a function f(x) is maximal (Fig. 46). Under a smooth change of the function the optimal solution changes with a jump from one of the two competing maxima (A) to the other (B).

Keywords

Normal Form Decision Theory Maximum Function Minimum Function Catastrophe Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliographical Comments

  1. L. N. Bryzgalova: Singularities of the maximum of a parametrically dependent function. Funkts. Anal. Prilozh. 11:1 (1977), 59–60 (English translation: Funct. Anal. Appl. 11 (1977), 49-51).MathSciNetzbMATHGoogle Scholar
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  3. V. A. Vasil’ev: Asymptotic exponential integrals, Newton’s diagram, and the classification of minimal points. Funkts. Anal. Prilozh. 11:3 (1977), 1–11 (English translation: Funct. Anal. Appl. 11 (1977), 163-172).Google Scholar
  4. V. I. Matov: The topological classification of germs of the maximum and minimax functions of a family of functions in general position. Usp. Mat. Nauk 37:4 (1982), 129–130 (English translation: Russ. Math. Surv. 37:4(1982), 127-128).MathSciNetGoogle Scholar
  5. V. I. Matov: Ellipticity domains for generic families of homogeneous polynomials and extremum functions (in Russian). Funkts. Anal. Prilozh. 19:2 (1985), 26–36 (English translation to appear in Funct. Anal. Appl. 19 (1985)).MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Vladimir Igorevich Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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