Related papers: Boundary optimization for rough sets

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing…

We consider the problem of automatically proving resource bounds. That is, we study how to prove that an integer-valued resource variable is bounded by a given program expression. Automatic resource-bound analysis has recently received…

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a…

In this paper, we generalize the chance optimization problems and introduce constrained volume optimization where enables us to obtain convex formulation for challenging problems in systems and control. We show that many different problems…

Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can…

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model…

Set partitioning is a key component of many algorithms in machine learning, signal processing, and communications. In general, the problem of finding a partition that minimizes a given impurity (loss function) is NP-hard. As such, there…

This article aims to introduce the paradigm of distributional robustness from the field of convex optimization to tackle optimal design problems under uncertainty. We consider realistic situations where the physical model, and thereby the…

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint…

Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence…

We initiate the study of the algorithmic problem of certifying lower bounds on the discrepancy of random matrices: given an input matrix $A \in \mathbb{R}^{m \times n}$, output a value that is a lower bound on $\mathsf{disc}(A) = \min_{x…

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the…

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and…

This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the…

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…

Rapid growth in the field of quantitative digital image analysis is paving the way for researchers to make precise measurements about objects in an image. To compute quantities from the image such as the density of compressed materials or…

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate…

When randomized ensembles such as bagging or random forests are used for binary classification, the prediction error of the ensemble tends to decrease and stabilize as the number of classifiers increases. However, the precise relationship…

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…