''We will say that the members of a collectivity enjoy the

Though it does not explicitly refer to any optimization context, this sentence is widely recognized as the very root of what nowadays is known as

The core of this dissertation is devoted to the discussion of optimality conditions for nonlinear vector optimization problems. Much work has been done on this topic, starting from the pioneering papers of Da Cuna and Polak, Borwein, Lin, Censor, Ben-Israel et al. and many others followed. The main focus has generally been on first order optimality conditions both for problems with smooth and nonsmooth functions or on saddlepoint type criteria; on the contrary, just a few papers developed a second order analysis. Another issue that seems to have not been deeply investigated is Lagrangian regularity, that deals with those conditions which guarantee multipliers rules to hold with nonzero multipliers corresponding to all or at least one of the objective functions. In the case of classical nonlinear optimization such conditions generally involve just the constraint and therefore they are usually referred to as constraint qualifications; on the contrary, conditions involving also the objective functions are often needed in the case of vector optimization.

A large part of Chapter 2 is devoted to the analysis of regularity conditions both for first and second order multipliers rules. In particular, not only very weak ones are presented but also those regularity conditions, which guarantee additional properties of multipliers such as boundedness and uniquess, are analysed. It is worth stressing that the focus of that chapter is on first and second order optimality criteria both for the classical problems, where the constraint is described by inequalities and equalities, and for those problems where the constraint is given just as a set. Actually, necessary optimality criteria are first developed for the latter case, exploiting suitable approximations of the constraint; these results allow to deduce multipliers rules and analyse regularity conditions for the former case quite easily. This procedure does not work for sufficient criteria and the relationships between the criteria developed for the two cases are investigated: quite surprisingly, the second order ones are completely unrelated.

Since the results of Chapter 2 require that functions are smooth, precisely differentiable or twice differentiable, Chapter 3 is devoted to optimality criteria for problems involving nonsmooth functions. Two different approaches are considered and compared: the first one relies on the components of vector functions so that well-known nonsmooth analysis tools for the real--valued case can be exploited; the second one is based on set--valued directional derivatives, which can be introduced considering suitable limits of vector incremental ratios without relying on components. The latter allows also to recover tools for vector functions such as generalized Jacobians and Hessians as particular cases and furthermore it provides sharper results than the former.

Finally, Chapter 4 deals with optimality criteria based on the saddlepoints of suitable Lagrangian functions. Though this issue has been deeply investigated, vector Lagrangians with multipliers associated to the objective functions have generally been disregarded. On the contrary, they allow to achieve stronger results and to recover those based on the saddlepoint of real Lagrangians as particular cases.

The thesis was sent out for evaluation on October, 2002. The referees have been Prof. Angelo Guerraggio (Università dell'Insubria) and Prof. Kathrin Klamroth (Universität Erlangen-Nürnberg). The thesis has been successfully defended on September 19, 2003. The committee was chaired by Prof. Franco Giannessi (Università di Pisa) and composed by Prof. Massimo Pappalardo (Thesis advisor), Prof. Fioravante Patrone (Università di Genova), Proff. Angelo Guerraggio and Kathrin Klamroth.

Download the .pdf file (1.2M) of the thesis: