Argument for concavity: Definition looks more like concavity -non-increasing discrete derivatives. Argument for convexity: Submodularity seems to be more useful for minimization than maximization.

A number of interesting functions arising in combinatorial optimization turn out to be submodular. Linear functions: A function f : 2N ! R is linear if f(A) = P i2A wi for some weights. w : N ! R. Such functions are also referred to as additive or modular. If wi 0 for all i 2 N, then f is also monotone. 0, is monotone submodular.

submodular functions often appear as objective functions of machine learning tasks such as sensor placement, document summarization or active learning !simple algorithms such as Greedy or Local Search work well.

the minimum of submodular functions can be found in polynomial time, and the maximum can be constant-factor approximated in low-order polynomial time. Submodular functions share properties in...

In this paper, we describe two \greedy" approaches to the problem of sub-modular maximization. As we will show below, maximizing a submodular func-tion is provably hard in a strong sense; nevertheless, simple greedy algorithms provide approximations to optimal solutions in many cases of practical signif-icance.

Submodular function optimization is a fundamental tool in modeling complex interactions in machine learning and graph mining problems. We propose to study constrained submodular optimization to improve the current state of the art.

Optimizing submodular set functions: discrete optimization via continuous optimization • extensions via expectations • convex and partially concave Further connections: • Submodularity more generally: continuous optimization via discrete optimization • Negative dependence and stable polynomials

2021年11月7日 · Our main interest today is to give a polynomial time algorithm for minimizing an arbitrary submodular function. Even though submodular function is a discrete object, we will be able to minimize it by reduction to minimizing a continuous convex function over a convex set.

Submodular/supermodular functions are weak analogues to convex/concave functions (in no particular order!) Nonnegative: f(A) ≥ 0 for all S ⊆ X Normalized: f(∅) = 0. A ⊆ B ⊆ X and i 6∈B. A set function f : 2X → R is supermodular if and only if −f is submodular. for all A ⊆ B ⊆ X and i 6∈B.

2 天之前 · Submodular Optimization. Submodular optimization is a powerful framework for solving combinatorial optimization problems that exhibit diminishing returns (Nemhauser et al., 1978; Feige & Goemans, 1995; Cornuejols et al., 1977).In a monotone setting, adding more elements to a solution always increases its utility, but at a decreasing rate.