Directional Differentiability of a Continual Maximum Function of Quasidifferentiable Functions

Demyanov, V.F. & Zabrodin, I.S. (1983). Directional Differentiability of a Continual Maximum Function of Quasidifferentiable Functions. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-83-058

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Abstract

Much recent work in optimization theory has been concerned with the problems caused by nondifferentiability. Some of these problems have now been at least partially overcome by the definition of a new class of nondifferentiable functions called quasidifferentiable functions, and the extension of classical differential calculus to deal with this class of functions. This has led to increased theoretical research in the properties of quasidifferentiable functions and their behavior under different conditions.

In this paper, the problem of the directional differentiability of a maximum function over a continual set of quasidifferentiable functions is discussed. It is shown that, in general, the operation of taking the "continual" maximum (or minimum) leads to a function which is itself not necessarily quasidifferentiable.

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