In this paper, we give notions of well posedness for a vector optimization problem and a vector variational inequality of the differential type. First, the basic properties of well-posed vector ...optimization problems are studied and the case of C-quasiconvex problems is explored. Further, we investigate the links between the well posedness of a vector optimization problem and of a vector variational inequality. We show that, under the convexity of the objective function f, the two notions coincide. These results extend properties which are well known in scalar optimization. PUBLICATION ABSTRACT
The present paper studies the following constrained vector optimization problem: min
C
f
(
x
),
g
(
x
)∈−
K
,
h
(
x
)=0, where
f
:ℝ
n
→ℝ
m
,
g
:ℝ
n
→ℝ
p
and
h
:ℝ
n
→ℝ
q
are locally Lipschitz ...functions and
C
⊂ℝ
m
,
K
⊂ℝ
p
are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point
x
0
to be a
w
-minimizer (weakly efficient point) or an
i
-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.
The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems.
The origins of the notion of quasi-concave function are considered, with special interest in some work by John von Neumann, Bruno de Finetti, and W. Fenchel. The development of such pioneering ...studies subsequently led to a whole field of research, known as “generalized convexity.” The different styles of the three authors and the various motivations for introducing quasi-concavity are compared, without losing sight of economic applications characteristic of the whole field of generalized convexity.
Il lavoro considera le origini della nozione di funzione quasi-concava, con particolare riguardo ad alcuni scritti di John von Neumann, di Bruno de Finetti, e di W. Fenchel. Lo sviluppo di tali studi pionieristici ha successivamente consentito lo sviluppo di un intero campo di ricerca detto “convessità generalizzata.” I diversi stili dei tre autori e le differenti motivazioni che hanno portato all'introduzione della quasi-concavità vengono confrontati, senza perdere di vista il riferimento alle applicazioni economiche che costituiscono una caratteristica in tema di convessità generalizzata.
Vite matematiche Bartocci, C; Betti, R; Guerraggio, A ...
2008, 2007, 2007-05-04
eBook
Lo scibile matematico si espande a un ritmo vertiginoso. Nel corso degli ultimi cinquant'anni sono stati dimostrati più teoremi che nei precedenti millenni della storia umana. Per illustrare la ...ricchezza della matematica del Novecento, il presente volume porta sulla ribalta alcuni dei protagonisti di questa straordinaria impresa intellettuale, che ha messo a nostra disposizione nuovi e potenti strumenti per indagare la realtà che ci circonda.Presentando matematici famosi accanto ad altri meno noti al grande pubblico da Hilbert a Gödel, da Turing a Nash, da De Giorgi a Wiles i ritratti raccolti in questo volume ci presentano personaggi dal forte carisma personale, dai vasti interessi culturali, appassionati nel difendere limportanza delle proprie ricerche, sensibili alla bellezza, attenti ai problemi sociali e politici del loro tempo.Ne risulta un affresco che documenta la centralità della matematica nella cultura, non solo scientifica ma anche filosofica, artistica e letteraria, del nostro tempo, in un continuo gioco di scambi e di rimandi, di corrispondenze e di suggestioni.
"This book describes Italian mathematics in the period between the two World Wars. We analyze its development by focusing on both the interior and the external influences. Italian mathematics in that ...period was shaped by a colorful array of strong personalities who concentrated their efforts on a select number of fields and won international recognition and respect in an incredibly short time. Consequently, Italy was considered a third ""mathematical power"" after France and Germany, and qualified Italian universities became indispensable stops on the ""tour"", organized for the improvement of young foreign mathematicians. At that time, Italy was also dominated by a fascist regime. This political situation and the social and academic structure of Italian society are included in the analysis as influences external to mathematics itself. The authors have provided a fascinating study of a most difficult time in the history of the world and of mathematics."
The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order ...necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.
On general vector quasi-optimization problems GUERRAGGIO, Angelo; NGUYEN XUAN TAN
Mathematical methods of operations research (Heidelberg, Germany),
06/2002, Volume:
55, Issue:
3
Journal Article
Peer reviewed
Vector general quasi-optimization problems are formulated and some sufficient conditions on the existence of solutions for these problems are shown. These concern the existence of solutions, the ...stability and the structure of solution set of general vector problems. As special case, we obtain results on the existence of solutions of vector quasi-optimization problem. Some relationships between these problems and others, as vector quasi-equilibrium problems, quasi-variational inequalities, complementarity problems,..., are shown. From these we extend some well-known results obtained by Blum and Oettli 4, Park 21, Chan and Pang 6, Parida and Sen 20, Browder and Minty 18, Fan 8, etc.
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