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Journal of Convex Analysis, Vol. 5, No. 1, pp. 153-170 (1998)
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Vertical Developments of a Convex Function

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Francois Oustry

INRIA, Domaine de Voluceau-Rocquencourt, B.P. 105, 78153 Le Chesnay, France, oustry@inria.fr

**Abstract:** In this paper we compare two different approaches to analyse the second-order behaviour of a convex function. The first one is classical, we call it the * horizontal approach*; the second one is more recent, it is the * vertical approach*. We prove equivalences between horizontal and vertical growth conditions. Then we derive well-known directional results. Finally we show that the vertical approach is particularly interesting to get more than a first-order (and more than directional) analysis of the maximum eigenvalue function.

**Keywords:** Convex analysis, second-order derivative, approximate subdifferential, semidefinite programming

**Full text of the article:**

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