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Dini derivatives in optimization. I. (English) Zbl 0871.49019

In the two chapters of the paper, the authors provide an interesting survey of the properties of the different Dini derivatives of functions with one and with several variables.
For the one-dimensional case the upper/lower right/left Dini derivatives \(D^+f(t)\), \(D_+f(t)\), \(D^-f(t)\), \(D_-f(t)\) of the function \(f\) are introduced and discussed. After a short comparison of all the derivations and the formulation of simple calculus rules some relations to continuity and differentiability are pointed out. In a large scale generalized mean value theorems are formulated with and without (semi) continuity assumptions. The results are used for the characterization of monotonicity and Lipschitz continuity in terms of Dini derivatives and for the description of first and second order optimality conditions.
Chapter 2 is concerned with the discussion of the different radial Dini derivatives and the Dini-Hadamard derivatives of functions of several variables. After pointing out the relations to Gâteaux and Fréchet differentiability, the authors provide some optimality conditions for free and constrained optimization problems and some calculus rules for the calculation of the Dini derivatives of max-functions.
The paper is completed by a second part in which the authors use the results for the discussion of generalized convexity and the derivation of general optimality conditions.

MSC:

49J52 Nonsmooth analysis
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26B05 Continuity and differentiation questions
49K05 Optimality conditions for free problems in one independent variable
49J50 Fréchet and Gateaux differentiability in optimization
49K10 Optimality conditions for free problems in two or more independent variables
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