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Exhausters of a positively homogeneous function. (English) Zbl 0954.90050
Summary: Notions of upper exhauster and lower exhauster of a positively homogeneous (of the first degree) function $$h: \mathbb{R}^n\to \mathbb{R}$$ are introduced. They are linked to exhaustive families of upper convex and lower concave approximations of the function $$h$$. The pair of an upper exhauster and a lower exhauster is called a biexhauster of $$h$$. A calculus for biexhausters is described (in particular, a composition theorem is formulated). The problem of minimality of exhausters is stated. Necessary and sufficient conditions for a constrained minimum and a constrained maximum of a directionally differentiable function $$f: \mathbb{R}^n\to \mathbb{R}$$ are formulated in terms of exhausters of the directional derivative of $$f$$. In general, they are described by means of exhausters of the Hadamard upper and lower directional derivatives of the function $$f$$. To formulate conditions for a minimum, an upper exhauster is employed while conditions for a maximum are formulated via a lower exhauster of the respective directional derivative (the Hadamard lower derivative for a minimum and the Hadamard upper derivative for a maximum).
If a point $$x_0$$ is not stationary then directions of steepest ascent and descent can also be calculated by means of exhausters.

##### MSC:
 90C30 Nonlinear programming
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