## On generalized second-order derivatives and Taylor expansions in nonsmooth optimization.(English)Zbl 0801.49016

The authors study generalized directional derivatives for obtaining Taylor expansions. The Cominetti and Correa formula for second order directional derivatives is used $f^ \infty (x;u,v)= \lim \sup _{\substack{ y\to x\\ s,t\downarrow 0 }}\;\textstyle {{1\over {ts}}} \bigl\{ f(y+ tu+ sv)- f(y+ tu)- f(y+ sv)+ f(y) \bigr\}.$ On this basis the authors represent $$f^ \infty (x; u,v)$$ in the form of upper limit of the rates of changes of the Dini-directional derivatives. Then they apply their results to large class of functions not covered by classical literature. Applications to optimization theory are presented in paragraph 6.

### MSC:

 49J52 Nonsmooth analysis 49K27 Optimality conditions for problems in abstract spaces 49M37 Numerical methods based on nonlinear programming 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26E15 Calculus of functions on infinite-dimensional spaces
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