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Extended Monotropic Programming and Duality

Abstract

We consider the problem

$$\begin{array}{l@{\quad}l}\mbox{min}&\displaystyle\sum_{i=1}^{m}f_{i}(x_{i}),\\[12pt]\mbox{s.t.}&x\in S,\end{array}$$

where x i are multidimensional subvectors of x, f i are convex functions, and S is a subspace. Monotropic programming, extensively studied by Rockafellar, is the special case where the subvectors x i are the scalar components of x. We show a strong duality result that parallels Rockafellar’s result for monotropic programming, and contains other known and new results as special cases. The proof is based on the use of ε-subdifferentials and the ε-descent method, which is used here as an analytical vehicle.

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Correspondence to D. P. Bertsekas.

Additional information

Communicated by P. Tseng.

Work partially supported by the National Science Foundation Grant No. CCR-9731273.

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Bertsekas, D.P. Extended Monotropic Programming and Duality. J Optim Theory Appl 139, 209–225 (2008). https://doi.org/10.1007/s10957-008-9393-3

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  • DOI: https://doi.org/10.1007/s10957-008-9393-3

Keywords

  • Monotropic
  • Duality
  • ε-subdifferential
  • ε-descent