Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have delved into the maximization of non-monotone DR-submodular functions over general (not necessarily down-closed) convex set constraints. Up to this point, these works have all used the minimum $\ell_\infty$ norm of any feasible solution as a parameter. Unfortunately, a recent hardness result due to Mualem \& Feldman~\cite{mualem2023resolving} shows that this approach cannot yield a smooth interpolation between down-closed and non-down-closed constraints. In this work, we suggest novel offline and online algorithms that provably provide such an interpolation based on a natural decomposition of the convex body constraint into two distinct convex bodies: a down-closed convex body and a general convex body. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.

在非凸优化的领域中，DR-submodular函数的优化在最近越来越重要，一些最近的工作探讨了在一般（不一定是下闭的）凸集约束下非单调DR-submodular函数的最大化，但之前的方法使用最小的L∞范数作为参数，而Mualem和Feldman的研究结果表明这种方法无法在下闭和非下闭约束之间进行平滑插值。本文提出了基于凸体约束的自然分解方法，通过提供下闭凸体和一般凸体两个不同凸体的插值，我们还通过三个离线应用和两个在线应用实证了我们所提算法的优越性。

将一般和闭凸集连接起来：子模最大化中的桥梁