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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.

90C30 Nonlinear programming
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