On Proper Separation Theorems by Means of the Quasi-Relative Interior with Applications

Tijani Amahroq (), Hassan Khatite () and Abdessamad Oussarhan
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Tijani Amahroq: LAMAI Laboratory, Faculty of Sciences and Techniques, Cadi Ayyad University, B. P. 549, Marrakech, Morocco
Hassan Khatite: LAMAI Laboratory, Faculty of Sciences and Techniques, Cadi Ayyad University, B. P. 549, Marrakech, Morocco
Abdessamad Oussarhan: LIMATI Laboratory, Polydisciplinary Faculty, Sultan Moulay Slimane University, B. P. 592, Beni Mellal, Morocco

Asia-Pacific Journal of Operational Research (APJOR), 2024, vol. 41, issue 05, 1-23

Abstract: In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

Keywords: Proper separation theorem; quasi-relative interior; pseudo-relative interior; quasi-interior; vector optimization; Karush–Kuhn–Tucker multipliers; Fritz-John multipliers; optimality conditions (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S021759592350032X

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