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Second-order moderate derivatives. (English) Zbl 0810.49017
For a real-valued function $$f$$ defined on a normed vector space, this paper introduces two kinds of second-order directional derivatives: $f^{}(x, u,v)= \sup_{w,z\in X} \limsup_{s,t\to 0_ +} \Delta^ 2_{s,t} f(x,u,v,w,z),$ where $$\Delta^ 2_{s,t} f(x,u,v,w,z)= {1\over st}[f(y+ su+ tv)- f(y+ su)- f(y+ tv)- f(y)]$$, with $$y= x+ sw+ tz$$ and $f^{\square\square}(x,u,v)= \sup_{w\in X} \limsup_{t\to 0_ +} \textstyle{{1\over t}} [f^ \square (x+ tw+ tv,u)- f^ \square(x+ tw,u)],$ where $$f^ \square(x,v)= \sup_{w\in X} \limsup_{t\to 0_ +} {1\over t} [f(x+ tw+ tv)- f(x+ tw)]$$. It is shown in this paper that if $$f$$ is Gâteaux differentiable and -semi- differentiable, that is, $f''(x,u,v)= \lim_{s,t\to 0_ +}\textstyle{{1\over st}} [f(x+ su+ tv)- f(x+ su)- f(x+ tv)+ f(x)],$ exists, then $f^{}(x, u,v)= f^{\square\square}(x,u,v)= f''(x, u,v).\tag{1}$ It would be interesting to find a more general class of functions where the first equality in (1) still holds. Calculus rules, such as chain rules and Taylor expansions, are established using these generalized second-order directional derivatives. Applications are given to derive optimality conditions of optimization problems.

##### MSC:
 49J52 Nonsmooth analysis
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##### References:
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