STATIONARITY AND REGULARITY OF SET SYSTEMS ALEXANDER KRUGER Dedicated to R. T. Rockafellar on his 70th birthday Abstract. Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle ) are presented for the case of an Asplund space. 1. Introduction Starting with the pioneering work by Dubovitskii and Milyutin [8] it is quite natural when dealing with optimality conditions to reformulate optimality in the original optimization problem as a (some kind of) extremal behaviour of a certain system of sets. An easy example is a problem of unconditional minimization of a real-valued function ϕ : X → R. If x ◦ ∈ X one can consider the sets Ω 1 = epi ϕ = {(x, μ) ∈ X × R : ϕ(x) ≤ μ} (the epigraph of ϕ) and Ω 2 = X ×{μ : μ ≤ ϕ(x ◦ )} (the lower halfspace). The local optimality of x ◦ is then equivalent to the condition Ω 1 ∩ int Ω 2 ∩ B ρ (x ◦ )= ∅ for some ρ> 0. Considering set systems is a rather general scheme of investigating optimization problems. Any set of “extremality” conditions leads to some optimality conditions for the original problem. When the sets are convex (or admit some convex approximations) extremality conditions are given by the separation theorem. In the general case a nonconvex separation theorem (the generalized Euler equation ) was proved in [22]. By now it is generally referred to as the Extremal principle (see [24, 33]) and has numerous applications to optimization, calculus and economics. A different (but in a sense equivalent) scheme of investigating nonconvex set systems was developed in [4]. Any necessary optimality conditions characterize in the nonconvex case not only opti- mal solutions but some broader set of stationary points which can also be of interest. The stationarity notion corresponding to the extremal principle conditions, namely weak station- arity, was investigated in [21]. Introducing weak stationarity made possible to reformulate the (Extended) extremal principle as a necessary and sufficient condition. 2000 Mathematics Subject Classification. 90C46, 90C48, 49K27; Secondary: 58C20, 58E30. Key words and phrases. nonsmooth analysis, normal cone, optimality, extremality, stationarity, regular- ity, set-valued mapping, Asplund space. 1