Reviews: Efficient Algorithms for Non-convex Isotonic Regression through Submodular Optimization

In this paper the authors consider the problem of minimizing a continuous submodular function subject to ordering (isotonic) constraints. They first show that the problem can be solved if we first discretize it (per coordinate, not in [0,1] n), and then solve the resulting discrete optimization problem using convex optimization. The fact that the problem is solvable in polynomial time is of course not surprising, because, as pointed out by the authors in lines 29-36, we can add a penalty to the objective that will implicitly enforce the constraints. However, this can significantly increase the Lipschitz constant of the objective, and that is why the authors take on an alternative approach. First, they prove that seen in the space of measures, the isotonic constraints correspond to dominating inequalities of the CDFs, which I guess is an intuitive result given the results known for the unconstrained case.