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In this paper we introduce the regenerator location problem (RLP), which deals with a constraint on the geographical extent of transmission in optical networks. Specifically, an optical signal can only travel a maximum distance of d max before its quality deteriorates to the point that it must be regenerated by installing regenerators at nodes of the network. As the cost of a regenerator is high we wish to deploy as few regenerators as possible in the network, while ensuring all nodes can communicate with each other. We show that the RLP is NP-Complete. We then devise three heuristics for the RLP. We show how to represent the RLP as a max leaf spanning tree problem (MLSTP) on a transformed graph. Using this fact we model the RLP as a Steiner arborescence problem (SAP) with a unit degree constraint on the root node. We also devise a branch-and-cut procedure to the directed cut formulation for the SAP problem. In our computational results over 740 test instances, the heuristic procedures obtained the optimal solution in 454 instances, while the branch-and-cut procedure obtained the optimal solution in 536 instances. These results indicate the quality of the heuristic solutions are quite good, and the branch-and-cut approach is viable for the optimal solution of problems with up to 100 nodes. Our approaches are also directly applicable to the MLSTP indicating that both the heuristics and branch-and-cut approach are viable options for the MLSTP.

T he uncapacitated facility location (UFL) problem is one of the most famous and most studied problems in the operations research literature. Given a set of potential facility locations and a set of customers, the goal is to find a subset of facility locations to open and to allocate each customer to open facilities so that the facility opening plus customer allocation costs are minimized. In our setting, for each customer the allocation cost is assumed to be a linear or separable convex quadratic function. Motivated by recent UFL applications in business analytics, we revise approaches that work on a projected decision space and hence are intrinsically more scalable for large-scale input data. Our working hypothesis is that many of the exact (decomposition) approaches that were proposed decades ago and discarded soon after need to be redesigned to take advantage of the new hardware and software technologies. To this end, we "thin out" the classical models from the literature and use (generalized) Benders cuts to replace a huge number of allocation variables by a small number of continuous variables that model the customer allocation cost directly. Our results show that Benders decomposition allows for a significant boost in the performance of a mixed-integer programming solver. We report the optimal solution of a large set of previously unsolved benchmark instances widely used in the available literature. In particular, dramatic speedups are achieved for UFL problems with separable quadratic allocation costs, which turn out to be much easier than their linear counterparts when our approach is used.

The Steiner tree problem is a challenging NP-hard problem. Many hard instances of this problem are publicly available, that are still unsolved by state-of-theart branch-and-cut codes. A typical strategy to attack these instances is to enrich the polyhedral description of the problem, and/or to implement more and more sophisticated separation procedures and branching strategies. In this paper we investigate the opposite viewpoint, and try to make the solution method as simple as possible while working on the modeling side. Our working hypothesis is that the extreme hardness of

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, optimal expansion of gas networks, critical infrastructure defense, and machine learning. In this paper, we present a new general purpose branch-and-cut framework for the exact solution of mixed-integer bilevel linear programs (MIBLP), which constitute a very significant subfamily of bilevel optimization problems. Our framework introduces several new classes of valid inequalities to speed-up the solver, along with a very effective bilevel-specific preprocessing procedure. A very extensive computational study is presented, where we evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature-this is by far the most extensive computational analysis ever performed for exact MIBLP solvers. Our new algorithm consistently outperforms (often by a large margin) all alternative state-of-the-art methods from the literature, including methods which exploit problem specific information for special instance classes. In particular, it allows to solve to optimality more than 300 instances previously unsolved instances from literature.

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%.

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Abstract. The simple plant location problem (SPLP) is considered and a genetic algorithm is proposed to solve this problem. By using the developed algorithm it is possible to solve SPLP with more than 1000 facility sites and customers. Computational results are presented and compared to dual based algorithms.