A Unified Separation Theorem for Closed Sets in a Banach Space and Optimality Conditions for Vector Optimization

Abstract

Abstract. Using the technique of variational analysis and in terms of normal cones, we establish unified separation results for finitely many closed (not necessarily convex) sets in Banach spaces, which not only cover the existing nonconvex separation results and a classical convex separation theorem, but also recapture the approximate projection theorem. With help of the separation result for closed sets, we provide necessary and sufficient conditions for approximate Pareto solutions of constrained vector optimization problems. In particular, we extend some basic optimality results for approximate solutions of numerical optimization problems to the vector optimization setting.

About Extensions of the Extremal Principle

“…Thus, Theorem 4 is a special case of Theorem 3(i). On the other hand, as demonstrated in [35,36], Theorem 4 (as well as its version formulated above as Theorem 4 ) is sufficient for many important applications. Next we show that Theorem 4 implies the nonlocal version of the extremal principle.…”

“…Zheng and Ng, 2011) Suppose X is an Asplund space, A, B ⊂ X are closed, A ∩ B = ∅. If points a ∈ A and b ∈ B satisfy condition (10) with some ε > 0, then, for any λ > 0 and τ ∈ (0, 1), there exist points a ∈ A ∩ B λ (a), b ∈ B ∩ B λ (b) and a * ∈ X * such that…”

Extremality, Stationarity and Generalized Separation of Collections of Sets

“…Subdifferentiating a norm (in either X n−1 or X n ) at a nonzero point is a necessary step in the proofs of all existing versions of the extremal principle and its extensions, starting with the very first one in [12, Theorem 6.1], with conditions like (45) hidden in the proofs. In several statements in the rest of this section, following [31], we make such conditions exposed. (iii) Lemma 5.1 substitutes in the proof of Theorem 6.1 the conventional Ekeland variational principle.…”

“…Lemma 2.1. The statements have been further polished and analysed in a sequence of subsequent papers [28][29][30][31]. In particular, it was proved by Guoyin Li et al in [30,Theorem 3.1] that the conclusion of [27,Lemma 2.2] is actually equivalent to the Ekeland variational principle, and as such implies (the Banach space with Clarke normal cones version of) the extremal principle.…”

“…A partial comparison of the assumptions and conclusions in the two existing εextremality statements in [22,Theorem 3.1] and [31, Theorems 3.1 and 3.4] has been done recently in [32], where it was also noted that they are formulated using in a sense different languages and are in general incomparable. In the current paper, we formulate in Theorem 6.1 (and prove) a new general ε-extremality statement which exposes the core arguments and the role of the parameters in the conventional proofs of dual characterizations of the extremality and stationarity and, in particular, implies [22,Theorem 3.1] and [31,Theorems 3.1 and 3.4]. In view of the above-mentioned result by Guoyin Li et al, the conclusion of Theorem 6.1 is also equivalent to the Ekeland variational principle.…”