Dini derivatives in optimization — Part I

This paper, published in two parts, is mainly concerned with general properties of Dini derivatives of functions of one and several variables and with some applications of this topic to the study of generalized convexity and generalized optimality conditions for mathematical programming problems.In part I the basic definitions and properties are given, with reference both to functions of one real variable and to functions of several real variables. In this part special attention is given to the restatement of the basic theorems of the classical analysis to nondifferentiable functions, in terms of Dini derivatives.In part II we use these derivatives in order to define some classes of nondifferentiable generalized convex functions and the class of generalized upper quasidifferentiable functions. This part concludes with the development of optimality conditions for a nonsmooth programming problem, expressed in terms of the tools prevously introduced.RiassuntoIl presente lavoro, pubblicato in due parti, riguarda le principali proprietà dei numeri derivati di Dini (o derivate direzioni di Dini), sia di funzioni di una variablile che di più variabili, nonché alcune loro applicazioni allo studio della convessità generalizzata ed a problemi di ottimizzazione vincolata.Nella prima parte del lavoro si forniscono le definizioni e le proprietà fondamentali dei numeri derivati di Dini e vengono riformulati alcuni classici teoremi dell'an alisi, con riferimento a funzioni non differenziabili.Nella seconda parte tali derivate direzionali vengono applicate nello studio di alcune classi di funzioni convesse generalizzate non differenziabili e nell'ottenimento di condizioni di ottimalità per problemi (non differenziabili) di programmazione matematica.

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