Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling
The paper represents new results on first order necessary optimality conditions for local and global minima in set-valued optimization, multiplier rules included (Theorem 4.3). The application to economics with \(n\in\mathbb N\) customers and \(m\in \mathbb N\) producers in normed commodity spaces yields a generalized second fundamental theorem of welfare economics (multiplier rule, Theorem 5.2). The used main tools are variational analysis, generalized differentiation for multivalued mappings (generalized normal cones) and very general preference mappings. The transitivity of the ordering is replaced by a much more general local shift property, the so called asymptotic closedness of the level sets with respect to the generalized ordering. The paper starts with a very general extremal principle for partially sequentially normally compact finite set systems (cf. [\textit{B. S. Mordukhovich}, Variational analysis and generalized differentiation. I: Basic theory. II: Applications. Grundlehren der Mathematischen Wissenschaften 331/332. Berlin: Springer (2006; Zbl 1100.49002)]) (part II, Lemma 5.58) and apply this principle for deriving necessary conditions in set-valued optimization - vector optimization included (Theorem 3.3). They consider three kinds of local minimizers: (i) fully localized weak minimizers (generalization of weak minimizers in vector optimization), (ii) fully localized minimizers (similar to Pareto minimality) and (iii) fully localized strong minimizers which mainly base on different asymptotic closedness properties of the used level set mappings. Comparisons of the new concept to former results in set-valued and vector optimization are given. The proofs of the important results are given in detail. Unfortunately, not all used notations are introduced e. g. the locally closedness of sets (closedness with respect to the intersection with some closed ball). For global minima in this concept, the asymptotic closedness of the level sets is replaced weaker by a refined net demand qualification condition. The condition is too weak for the local analysis as it is shown in an example (Remark 1).
nonsmooth and multiobjective optimization
second fundamental theorem of welfare economics
refined net demand qualification condition
scientific article; zbMATH DE number 5990793