Stable and total fenchel duality for dc optimization problems in locally convex spaces

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

doi:10.1137/100789749

Stable and total fenchel duality for dc optimization problems in locally convex spaces

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

Research output: Journal article publication › Journal article › Academic research › peer-review

39 Citations (Scopus)

Abstract

We consider the DC (difference of two convex functions) optimization problem (P) inf xεX {(F1(x) - f2((x)) + (g 1((Ax) - g2((Ax))}, where f 1(, f 2(, g1(, and g2( are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, and A is a linear continuous operator from X to Y. Adopting different tactics, two types of the Fenchel dual problems of (P) are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of (P) are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.

Original language	English
Pages (from-to)	730-760
Number of pages	31
Journal	SIAM Journal on Optimization
Volume	21
Issue number	3
DOIs	https://doi.org/10.1137/100789749
Publication status	Published - 24 Oct 2011

Keywords

Difference of two convex functions programming
Strong Fenchel duality
Total Fenchel duality

ASJC Scopus subject areas

Software
Theoretical Computer Science

More information

10.1137/100789749

http://hdl.handle.net/10397/6098

Cite this

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AU - Fang, D. H.

AU - Li, C.

AU - Yang, Xiaoqi

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N2 - We consider the DC (difference of two convex functions) optimization problem (P) inf xεX {(F1(x) - f2((x)) + (g 1((Ax) - g2((Ax))}, where f 1(, f 2(, g1(, and g2( are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, and A is a linear continuous operator from X to Y. Adopting different tactics, two types of the Fenchel dual problems of (P) are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of (P) are provided. Sufficient and/or necessary conditions for the strong Fenchel duality, the stable Fenchel duality, and the stable total duality are derived.

AB - We consider the DC (difference of two convex functions) optimization problem (P) inf xεX {(F1(x) - f2((x)) + (g 1((Ax) - g2((Ax))}, where f 1(, f 2(, g1(, and g2( are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, and A is a linear continuous operator from X to Y. Adopting different tactics, two types of the Fenchel dual problems of (P) are given. By using the properties of the epigraph of the conjugate functions, some sufficient and necessary conditions for the weak duality of (P) 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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Stable and total fenchel duality for dc optimization problems in locally convex spaces

Abstract

Keywords

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