Abstract The quadratic linear ordering problem models a large number of applications in a variety of different domains. To solve it exactly, polyhedral methods have been proposed but their development is still in its beginning. Specifically, while it is evident that only a fraction of the triangle inequalities, which are most commonly known from the boolean quadric polytope, take part in a minimal linear description of the polytope associated with a canonical formulation of the quadratic linear ordering problem, it is broadly unclear to which of these inequalities this applies and how to distinguish them from the others. At the same time, these inequalities are essential to build strong polyhedral and semidefinite programming relaxations. Addressing these open questions and potentials, we reveal the desired combinatorial pattern that enables to identify the triangle inequalities which are facet-inducing and deduce a corresponding exact polynomial-time separation algorithm.

Abstract Clustering is one of the most fundamental tools in data science and machine learning, and k-means clustering is one of the most common of such methods. There is a variety of approximate algorithms for the k-means problem, but only a few methods compute the globally optimal solution, as it is in general NP-hard. In this paper, we consider the k-means problem for instances with low-dimensional data and formulate it as a structured concave assignment problem. This allows us to exploit the low-dimensional structure and solve the problem to global optimality for very large data sets with several clusters, complementing and outperforming state-of-the-art for this class of problems. The method builds on iteratively solving a small concave problem and a large linear programming or assignment problem. This gives a sequence of feasible solutions along with bounds, which we show converges to a zero optimality gap. The paper combines methods from global optimization to accelerate the procedure, and we provide numerical results on synthetic data and real-world data.