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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: \[ f^{[2]}(x, u,v)= \sup_{w,z\in X} \limsup_{s,t\to 0_ +} \Delta^ 2_{s,t} f(x,u,v,w,z), \] 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 [2]-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 \[ f^{[2]}(x, u,v)= f^{\square\square}(x,u,v)= f''(x, u,v).\tag{1} \] 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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