Arxiv:2104.13510V2 [Math.FA]
Fenchel-Rockafellar Theorem in Infinite Dimensions via Generalized Relative Interiors D. V. Cuong 1,2, B. S. Mordukhovich3, N. M. Nam4, G. Sandine5 Abstract. In this paper we provide further studies of the Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces. Key words. Relative interior, quasi-relative interior, intrinsic relative interior, quasi-regularity, Fenchel duality. AMS subject classifications. 49J52, 49J53, 90C31 1 Introduction Duality theory has a central role in optimization theory and its applications. From a primal optimization problem with the objective function defined in a primal space, a dual problem in the dual space is formulated with the hope that the new problem is easier to solve, while having a close relationship with the primal problem. Its important role in optimization makes the duality theory attractive for extensive research over the past few decades; see, e.g., [1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 25, 26, 28] and the references therein. Given a proper convex function f and a proper concave function g defined on Rn, the classical Fenchel-Rockafellar theorem asserts that inf f(x) g(x) x Rn = sup g (x∗) f ∗(x∗) x∗ Rn , (1.1) { − | ∈ } { ∗ − | ∈ } under the assumption that ri (dom (f)) ri (dom (g)) = ; see [27, Theorem 31.1].
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