An Extended Extremal Principle with Applications to Multiobjective Optimization

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We develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization-related problems. The main difference between the conventional extremal principle and the extended version developed below is that, instead of the translation of sets involved in the extremal systems, we allow deformations. The new version seems to be more flexible in various applications and covers, in particular, multiobjective optimization problemswith general preference relations. In this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games.