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2011•SIAM Journal on OptimizationRequires access

Stable and Total Fenchel Duality for DC Optimization Problems in Locally Convex Spaces

D. H. Fang, C. Li, Xiaoqi Yang

Open publisher page 33 citations

Abstract

We consider the DC (difference of two convex functions) optimization problem [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are proper convex functions defined on locally convex Hausdorff topological vector spaces [Formula: see text] and [Formula: see text], and [Formula: see text] is a linear continuous operator from [Formula: see text] to [Formula: see text]. Adopting different tactics, two types of the Fenchel dual problems of [Formula: see text] are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of [Formula: see text] are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.

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What this paper is about

We consider the DC (difference of two convex functions) optimization problem [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are proper convex functions defined on locally convex Hausdorff topological vector spaces [Formula: see text] and [Formula: see text], and [Formula: see text] is a linear continuous operator from [Formula: see text] to [Formula: see text]. Adopting different tactics, two types of the Fenchel dual problems of [Formula: see text] are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of [Formula: see text] are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.

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Available abstract

We consider the DC (difference of two convex functions) optimization problem [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are proper convex functions defined on locally convex Hausdorff topological vector spaces [Formula: see text] and [Formula: see text], and [Formula: see text] is a linear continuous operator from [Formula: see text] to [Formula: see text]. Adopting different tactics, two types of the Fenchel dual problems of [Formula: see text] are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of [Formula: see text] are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.

Key concepts: Mathematics, Perturbation function, Epigraph, Duality (order theory), Strong duality, Duality gap, Fenchel's duality theorem, Convex conjugate

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