## 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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### References:

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