Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization

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 L-infinity norm of any feasible solution as a parameter.Unfortunately, a recent hardness result due to Mualem and Feldman 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.