New Notions of Proper Efficiency in Set Optimization with the Set Criterion

Abstract

In this paper, we introduce new notions of proper efficiency in the sense of Henig for a set optimization problem by using the set criterion of solution. The relationships between them are studied. Also, we compare these concepts with the homologous ones given by considering the vector criterion. Finally, a Lagrange multiplier rule for Henig proper solutions of a set optimization problem with a cone constraint is obtained under convexity hypotheses. Illustrative examples are also given.

On proper minimality in set optimization

“…It follows that  ⊈ B + K 𝜌 . Otherwise, by (8) we have in particular that  ⊆ B + K , and we reach a contradiction. The proof continues, then, in the same way as in the proof of Theorem 1.…”

“…Probably, at a first sight, the most natural extension of the concept due to Henig to a set optimization problem is the H1-P minimality, based on the idea of replacing the cone K by a dilating cone C. The H1-P minimal points satisfy interesting properties. A first study of this notion can be found in [8], for a set-valued optimization problem and by considering the set criterion of solution (with the lower set less order relation).…”

“…To our knowledge, the only attempt that exists in the literature to extend the notion of proper minimality to the more general framework of set optimization was proposed in the very recent paper by Huerga et al [8], in which several notions of proper efficiency in the sense of Henig were defined and studied for a set-valued optimization problem, and by considering the set criterion of solution.…”

Minkowski Küme Sıralamaları için Minimallik ve Has Minimallik Üzerine

“…Vektör sıralamalarını küme değerli optimizasyona genişletilmesi ilk olarak Kuroiwa tarafından ortaya atıldı [10]. Literatürde küme optimizasyonu için has minimallik kavramı, Huerga ve arkadaşlarının yakın tarihli makalesinde önerildi [11]. Bu çalışmada, Emrah ve arkadaşlarının [12] tanımladığı küme sıralamaları ile işlem yapıldı.…”

“…Bulgular kısmında ise 𝑚 1 maksimallik ve 𝑚 1 minimallik kavramları ile ilgili önerme ve bu önermenin sonucu verildi. Huerga ve arkadaşlarının [13] makalesinde de küme optimizasyonundaki has minimallikler üzerine yeni çalışmalar verildi. 𝑚 1 has-minimal ve 𝑚 1 has-maksimal tanımları yapıldı.…”