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.

翻译：近年来，DR-子模函数优化在非凸优化领域取得了显著进展，其重要性日益凸显。受现实场景启发，近期研究开始探索在一般（非必然下闭）凸集约束下非单调DR-子模函数的最大化问题。迄今为止，这些工作均将可行解的最小$\ell_\infty$范数作为参数。遗憾的是，Mualem与Feldman~\cite{mualem2023resolving}近期的一项难解性结果表明，该方法无法在下闭约束与非下闭约束之间实现平滑插值。本研究提出新颖的离线与在线算法，通过将凸体约束自然分解为下闭凸体与一般凸体两部分，可证明实现此类插值。我们还通过三个离线应用与两个在线应用的实证实验，验证了所提算法的优越性。