The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function. Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic constraints on discrete variables. In a recent paper Atamtürk and Narayanan [6] study the lower level set of a non-decreasing submodular function. In this complementary paper we...

Techniques based on minimal graph cuts have become a standard tool for solving combinatorial optimization problems arising in image processing and computer vision applications. These techniques can be used to minimize objective functions written as the sum of a set of unary and pairwise terms, provided that the objective function is submodular. This can be interpreted as minimizing the $l_1$-no...

We study minimum entropy submodular optimization, a common generalization of the minimum entropy set cover problem, studied earlier by Cardinal et al., and the submodular set cover problem (Wolsey [Wol82], Fujishige [BIKP01], etc). We give a general bound of the approximation performance of the greedy algorithm using an approach that can be interpreted in terms of a particular type of biased ne...

Subset selection plays an important role in the field of evolutionary multiobjective optimization (EMO). Especially, EMO algorithm with unbounded external archive (UEA), subset is essential post-processing procedure to select a prespecified number solutions as final result. In this article, we discuss efficiency greedy for hypervolume, inverted generational distance (IGD), and IGD plus (IGD+) i...

Parallel Test Paper Generation (k-TPG) is a biobjective distributed resource allocation problem, which aims to generate multiple similarly optimal test papers automatically according to multiple userspecified criteria. Generating high-quality parallel test papers is challenging due to its NP-hardness in maximizing the collective objective functions. In this paper, we propose a Collective Biobje...

Solving stochastic optimization problems under partial observability, where one needs to adaptively make decisions with uncertain outcomes, is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy...

Recurrent neural network (RNN)’s architecture is a key factor influencing its performance. We propose algorithms to optimize hidden sizes under running time constraint. We convert the discrete optimization into a subset selection problem. By novel transformations, the objective function becomes submodular and constraint becomes supermodular. A greedy algorithm with bounds is suggested to solve ...

Unsupervised rank aggregation on score-based permutations, which is widely used in many applications, has not been deeply explored yet. This work studies the use of submodular optimization for rank aggregation on score-based permutations in an unsupervised way. Specifically, we propose an unsupervised approach based on the Lovasz Bregman divergence for setting up linear structured convex and ne...

We investigate a mixed 0 − 1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalit...

In this paper, we propose an unifying view of several recently proposed structured sparsityinducing norms. We consider the situation of a model simultaneously (a) penalized by a setfunction defined on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in `p-norm. We show that the natural combinatorial optimization problems obtained may ...