Complete mixed integer linear programming formulations for modularity density based clustering

Modularity density maximization is a clustering method that improves some issues of the commonly-used modularity maximization approach. Recently, some Mixed-Integer Linear Programming (MILP) reformulations have been proposed in the literature for the modularity density maximization problem, but they require as input the solution of a set of auxiliary binary Non-Linear Programs (NLPs). These can become … Read more

Divisive heuristic for modularity density maximization

In this paper we consider a particular method of clustering for graphs, namely the modularity density maximization. We propose a hierarchical divisive heuristic which works by splitting recursively a cluster into two new clusters by maximizing the modularity density, and we derive four reformulations for the mathematical programming model used to obtain the optimal splitting. … Read more

MILP formulations for the modularity density maximization problem

Cluster analysis refers to finding subsets of vertices of a graph (called clusters) which are more likely to be joined pairwise than vertices in different clusters. In the last years this topic has been studied by many researchers, and several methods have been proposed. One of the most popular is to maximize the modularity, which … Read more

RBFOpt: an open-source library for black-box optimization with costly function evaluations

We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the … Read more

Reformulation of a model for hierarchical divisive graph modularity maximization

Finding clusters, or communities, in a graph, or network is a very important problem which arises in many domains. Several models were proposed for its solution. One of the most studied and exploited is the maximization of the so called modularity, which represents the sum over all communities of the fraction of edges within these … Read more

On the impact of symmetry-breaking constraints on spatial Branch-and-Bound for circle packing in a square

We study the problem of packing equal circles in a square from the mathematical programming point of view. We discuss different formulations, we analyse formulation symmetries, we propose some symmetry breaking constraints and show that not only do they tighten the convex relaxation bound, but they also ease the task of local NLP solution algorithms … Read more