zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Strict efficiency in set-valued optimization. (English) Zbl 1201.90179
The authors extend the notion of $\phi$-strict minimizer to set-valued maps, in such a way that the concept of a $\phi$-strict minimizer, presented in {\it E. M. Bednarczuk} [Optimization 53, No. 5--6, 455--474 (2004; Zbl 1153.90529)] for vector functions, as well as the notion of a minimizer of order one introduced for set-valued maps in {\it G. P. Crespi, I. Ginchev} and {\it M. Rocca} [Math. Methods Oper. Res. 63, No. 1, 87--106 (2006; Zbl 1103.90089)] are generalized in a unified manner. A structure theorem is proved for a vector-valued function which states that a point is a $\phi$-strict minimizer for a vector valued function $f$ over a set if and only if the point is a $\phi$-strict minimizer for a family of scalar functions and sets, each of these functions being the composition of $f$ with a positive, continuous and linear functional, and the family of sets, a covering of the initial set. Different kinds of strict minimizers for the set-valued problem are characterized through different kinds of strict minimizers for an associated scalarized problem. Finally, several optimality conditions are established for strict minimizers of order one. A characterization is given for a global minimizer through the radial derivative. Comparisons with other results are made, and some illustrative examples are provide.

90C29Multi-objective programming; goal programming
90C46Optimality conditions, duality
90C31Sensitivity, stability, parametric optimization
Full Text: DOI