Given an infinite family of extended real-valued functions $$f_{i}$$ f i , $$i\in I,$$ i ∈ I , and a family $${\mathcal {H}}$$ H of nonempty finite subsets of I , the $${\mathcal {H}}$$ H -partial robust sum of $$f_{i}$$ f i , $$i\in I,$$ i ∈ I , is the supremum, for $$J\in {\mathcal {H}},$$ J ∈ H , of the finite sums $$\sum _{j\in J}f_{j}$$ ∑ j ∈ 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 $$ {\mathcal {H}}$$ 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 $${\mathcal {H}}$$ H -partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function.

中文翻译：

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约束鲁棒求和优化问题的对偶性

给定一个无限扩展实值函数族 $$f_{i}$$ fi , $$i\in I,$$ i ∈ I ，以及一个非空族 $${\mathcal {H}}$$ H I 的有限子集，$$f_{i}$$ fi , $$i\in I,$$ i ∈ I 的 $${\mathcal {H}}$$ H - 部分稳健和是最高的，对于 $$J\in {\mathcal {H}},$$ J ∈ H ，有限和 $$\sum _{j\in J}f_{j}$$ ∑ j ∈ J fj 。这些无限和以自然的方式出现在位置问题和函数逼近问题中，包括作为特殊情况的众所周知的 sup 函数和所谓的稳健和函数，对应于集合 $$ {\mathcal {H }}$$ H 的所有非空有限子集 I ，其无约束最小化在三位作者之前的论文中进行了分析（ https://doi.org/10.1007/s11228-019-00515-2 和 https://doi .org/10.1007/s00245-019-09596-9）。在本文中，