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).
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Bibliographical Comments
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).
L. N. Bryzgalova: Maximum functions of a family of functions depending on parameters. Funkts. Anal. Prilozh. 12:1 (1978), 66–67 (English translation: Funct. Anal. Appl. 12 (1978), 50-51).
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).
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).
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)).
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© 1986 Springer-Verlag Berlin Heidelberg
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Arnold, V.I. (1986). Singularities in Optimization Problems: the Maximum Function. In: Catastrophe Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96937-9_10
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DOI: https://doi.org/10.1007/978-3-642-96937-9_10
Publisher Name: Springer, Berlin, Heidelberg
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