\textit{Submodular continuous functions} are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, I) for monotone submodular continuous functions with an additional diminishing returns property, we propose a \texttt{Frank-Wolfe} style algorithm with $(1-1/e)$-approximation, and sub-linear convergence rate; II) for general non-monotone submodular continuous functions, we propose a \texttt{DoubleGreedy} algorithm with $1/3$-approximation. Submodular continuous functions naturally find applications in various real-world settings, including influence and revenue maximization with continuous assignments, sensor energy management, multi-resolution data summarization, facility location, etc. Experimental results show that the proposed algorithms efficiently generate superior solutions in terms of empirical objectives compared to baseline algorithms.

本文介绍了一种弱DR属性，用于给出有关子模性的统一特征，证明了可以在近似保证的同时高效地最大化子模连续函数，为在一般下闭凸约束条件下最大化单调DR-子模连续函数和在盒约束条件下的非单调子模连续函数提供了算法，并探讨了其在不同实际应用中的应用性。

保证的非凸优化：连续域上的子模最大化