This article comprises the first theoretical and computational study on mixed integer programming (MIP) models for the connected facility location problem (ConFL). ConFL combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized.We model ConFL using seven compact and three mixed integer programming formulations of exponential size. We also show how to transform ConFL into the Steiner arborescence problem. A full hierarchy between the models is provided. For two exponential size models we develop a branch-and-cut algorithm. An extensive computational study is based on two benchmark sets of randomly generated instances with up to 1300 nodes and 115,000 edges. We empirically compare the presented models with respect to the quality of obtained bounds and the corresponding running time. We report optimal values for all but 16 instances for which the obtained gaps are below 0.6%.

G iven a set of customers, a set of potential facility locations, and some interconnection nodes, the goal of the connected facility location problem (ConFL) is to find the minimum-cost way of assigning each customer to exactly one open facility and connecting the open facilities via a Steiner tree. The sum of costs needed for building the Steiner tree, facility opening costs, and the assignment costs needs to be minimized. If the number of edges between a prespecified node (the so-called root) and each open facility is limited, we speak of the hop constrained facility location problem (HC ConFL). This problem is of importance in the design of data-management and telecommunication networks.In this article we provide the first theoretical and computational study for this new problem that has not been studied in the literature so far. We propose two disaggregation techniques that enable the modeling of HC ConFL: (i) as a directed (asymmetric) ConFL on layered graphs, or (ii) as the Steiner arborescence problem (SA) on layered graphs. This allows for usage of best-known mixed integer programming models for ConFL or SA to solve the corresponding hop constrained problem to optimality. In our polyhedral study, we compare the obtained models with respect to the quality of their linear programming lower bounds. These models are finally computationally compared in an extensive computational study on a set of publicly available benchmark instances. Optimal values are reported for instances with up to 1,300 nodes and 115,000 edges.

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Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all p-norms for p ≥ 1. We show that the minimal p-distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for p = 1, the maximum minimal distance approaches the 1-diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.

The Hamiltonian p-median problem consists of determining p disjoint cycles of minimum total cost covering all vertices of a graph. We present several new and existing models for this problem, provide a hierarchy with respect to the quality of the lower bounds yielded by their linear programming relaxations, and compare their computational performance on a set of benchmark instances. We conclude that three of the models are superior from a computational point of view, two of which are introduced in this article.