Extended Monotropic Programming and Duality

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 xi are multidimensional subvectors of x, fi are convex functions, and S is a subspace. Monotropic programming, extensively studied by Rockafellar, is the special case where the subvectors xi 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.