Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization
This addresses a bottleneck in non-convex optimization for scenarios requiring flexible constraints, though it is incremental relative to existing work.
The paper tackles the problem of maximizing non-monotone DR-submodular functions over general convex sets, where prior methods using the minimum ℓ∞ norm fail to smoothly interpolate between down-closed and non-down-closed constraints. It proposes novel offline and online algorithms based on decomposing the convex body, showing empirical superiority in five applications.
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.