Calculus Rules on the Approximate Second-Order Directional Derivative of a Convex Function

Abstract

Given a real-valued convex function f, the approximate second-order directional derivative $f''_\varepsilon (x_0 ;d,d)$ of f at $x_0 $ in the direction d is an object which is defined whenever the parameter $\varepsilon $ is chosen strictly positive. The aim of this work is to derive expressions of the approximate second-order directional derivative of a function f which has been constructed from other functions $f_i $ whose properties are better known; we address ourselves to the problem of calculating $f''_\varepsilon $, having the ${(f_i )}''_\eta $, $\eta > 0$ at our disposal. Calculus rules are given for the main functional operations preserving convexity: composition with an affine mapping, sum of functions, image of a function under a linear mapping, maximum of functions.