The extremal principle and its applications to optimization and economics (Q2776680)

This is a survey article centering around the extremal principle introduced and extensively studied by the author and his collaborators.NEWLINENEWLINENEWLINEThis principle, which is too involved to be formulated here, has originally been developed to derive necessary optimality conditions for nonsmooth problems of optimization and optimal control. The author: ``This approach does not involve any convex approximations and convex separation arguments\dots The essence of the extremal principle is to provide necessary conditions for set extremality in terms of suitable normal cones in dual spaces that are not generated by tangent approximations in primal spaces and may be nonconvex.''NEWLINENEWLINENEWLINEA version of the extremal principle, the so-called fuzzy extremal principle, characterizes Asplund spaces among Banach spaces. It is also equivalent to a nonconvex analogue of the Bishop-Phelps theorem. The relationship of the fuzzy extremal principle to smooth as well as subdifferential versions of Ekeland's variational principle is also discussed.NEWLINENEWLINENEWLINEThe extremal principle is then applied in Asplund spaces to derive calculus rules for co-derivatives of multifunctions and in particular of subdifferentials. Moreover, it is applied to establish necessary optimality conditions in terms of normal cones and in terms of subdifferentials.NEWLINENEWLINENEWLINEFinally, applications of the extremal principle to nonconvex models of welfare economics with infinite-dimensional commodity spaces are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00048].