We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

Accepted:

DOI: 10.1051/cocv/2017024

Keywords: Generalized differentiation, positively homogeneous set-valued maps, linear openness, inverse mapping theorem, set-valued optimization, welfare economics

^{1}; Marcelin, Yvesner

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@article{COCV_2018__24_3_1059_0, author = {Geoffroy, Michel H. and Marcelin, Yvesner}, title = {A concept of inner prederivative for set-valued mappings and its applications}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1059--1074}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017024}, mrnumber = {3877193}, zbl = {1405.49011}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017024/} }

TY - JOUR AU - Geoffroy, Michel H. AU - Marcelin, Yvesner TI - A concept of inner prederivative for set-valued mappings and its applications JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1059 EP - 1074 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017024/ DO - 10.1051/cocv/2017024 LA - en ID - COCV_2018__24_3_1059_0 ER -

%0 Journal Article %A Geoffroy, Michel H. %A Marcelin, Yvesner %T A concept of inner prederivative for set-valued mappings and its applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1059-1074 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017024/ %R 10.1051/cocv/2017024 %G en %F COCV_2018__24_3_1059_0

Geoffroy, Michel H.; Marcelin, Yvesner. A concept of inner prederivative for set-valued mappings and its applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 1059-1074. doi : 10.1051/cocv/2017024. http://archive.numdam.org/articles/10.1051/cocv/2017024/

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