Active-set Methods for Submodular Optimization

We consider submodular optimization problems such as submodular function minimization (SFM) and quadratic problems regularized by the Lovasz extension; for cut functions, this corresponds respectively to graph cuts and total variation (TV) denoising. Given a submodular function with an SFM oracle, we propose a new active-set algorithm for total variation denoising, which is more flexible than existing ones; the algorithm may be seen as a local descent algorithm over ordered partitions with explicit convergence guarantees. For functions that decompose into the sum of two functions F1 and F2 with efficient SFM oracles, we propose a new active-set algorithm for total variation denoising (and hence for SFM by thresholding the solution at zero). This algorithm also optimizes over ordered partitions and improves over existing ones based on TV or SFM oracles for F1 and F2.