Duality for constrained robust sum optimization problems

Given an infinite family of extended real-valued functions $$f_{i}$$, $$i\in I,$$ and a family $${\mathcal {H}}$$ of nonempty finite subsets of I,  the $${\mathcal {H}}$$-partial robust sum of $$f_{i}$$, $$i\in I,$$ is the supremum, for $$J\in {\mathcal {H}},$$ of the finite sums $$\sum _{j\in J}f_{j}$$. These infinite sums arise in a natural way in location problems as well as in functional approximation problems, and include as particular cases the well-known sup function and the so-called robust sum function, corresponding to the set $$ {\mathcal {H}}$$ of all nonempty finite subsets of I,  whose unconstrained minimization was analyzed in previous papers of three of the authors (https://doi.org/10.1007/s11228-019-00515-2 and https://doi.org/10.1007/s00245-019-09596-9). In this paper, we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given $${\mathcal {H}}$$-partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function.