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Limiting behaviour of the approximate first order and second order directional derivatives for a convex function. (English) Zbl 0536.26007

Given \(\epsilon>0\), the following limit \[ f''_{\epsilon}(x_ 0;d,d)=\lim_{\alpha \to 0^+}\frac{f'_{\epsilon}(x_ 0+\alpha d;d)-f'_{\epsilon}(x_ 0;d)}{\alpha} \] is defined for all convex functions f, all \(x_ 0\) and all directions d. The aim of the paper is to study the behaviour of \(f''_{\epsilon}(x_ 0;d,d)\) when \(\epsilon\) goes to \(O^+\). It is proved that \(f''_{\epsilon}(x_ 0;d,d)\) does have a limit when \(\epsilon \to 0^+;\) more precisely: \[ \lim_{\epsilon \to 0^+}f''_{\epsilon}(x_ 0;d,d)=D''_ rf(x_ 0;d)=2V_ r''f(x_ 0;d), \] where \(D''_ rf(x_ 0;d)\) is the second right Dini derivative and \(V''_ rf(x_ 0;d)\) is the second right de la Vallée Poussin derivative. The following asymptotic development has also been derived: \[ f'_{\epsilon}(x_ 0;d)=f'(x_ 0;d)+[2\epsilon D''_ rf(x_ o;d)]^{1/2}+o(\epsilon^{1/2}). \] An example where such results are useful is the so-called discrete minimax problem, \(f=\max(f_ 1,...,f_ m),\) where the \(f_ i\) are \(C^ 2\) convex functions.

MSC:

26B05 Continuity and differentiation questions
90C25 Convex programming
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