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

  • Authors:

  • Chong Li;Donghui Fang;Genaro López;Marco A. López

  • Affiliations:

  • cli@zju.edu.cn;dh_fang@jsu.edu.cn;glopez@us.es;marco.antonio@ua.es

  • Venue:
  • SIAM Journal on Optimization
  • Year:
  • 2009

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Abstract

We consider the optimization problem $(P_{A})$ $\inf_{x\in X}\{f(x)+g(Ax)\}$ where $f$ and $g$ are proper convex functions defined on locally convex Hausdorff topological vector spaces $X$ and $Y$, respectively, and $A$ is a linear operator from $X$ to $Y$. By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of $(P_{A})$ are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived.