New models for the location of controversial facilities: A bilevel programming approach

Motivated by recent real-life applications in Location Theory in which the location decisions generate controversy, we propose a novel bilevel location model in which, on the one hand, there is a leader that chooses among a number of fixed potential locations which ones to establish. Next, on the second hand, there is one or several followers that, once the leader location facilities have been set, chooses his location points in a continuous framework. The leader’s goal is to maximize some proxy to the weighted distance to the follower’s location points, while the follower(s) aim is to locate his location points as close as possible to the leader ones. We develop the bilevel location model for one follower and for any polyhedral distance, and we extend it for several followers and any ℓp-norm, p ∈ Q, p ≥ 1. We prove the NP-hardness of the problem and propose different mixed integer linear programming formulations. Moreover, we develop alternative Benders decomposition algorithms for the problem. Finally, we report some computational results comparing the formulations and the Benders decompositions on a set of instances.Fonds de la Recherche Scientique - FNRSMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Variants on Dantzig-Wolfe Decomposition with Applications to Multistage Problems

The initial representation of an LP problem to which the Dantzig-Wolfe decomposition procedure is applied, is of the essence. We study this here, and, in particular, we consider two transformations of the problem, by introducing suitable linking rows and variables. We study the application of the Dantzig-Wolfe procedure to these new representations of the original problem and the relationship to previously proposed algorithms. Advantages and disadvantages from a computational viewpoint are discussed. Finally we develop a decomposition algorithm based upon these ideas for solving multistage staircase-structured LP problems

New models for the location of controversial facilities: A bilevel programming approach

International audienceMotivated by recent real-life applications in Location Theory in which the location decisions generate controversy, we propose a novel bilevel location model in which, on the one hand, there is a leader that chooses among a number of fixed potential locations which ones to establish. Next, on the second hand, there is one or several followers that, once the leader location facilities have been set, chooses his location points in a continuous framework. The leader's goal is to maximize some proxy to the weighted distance to the follower's location points, while the follower(s) aim is to locate his location points as close as possible to the leader ones. We develop the bilevel location model for one follower and for any polyhedral distance, and we extend it for several followers and any p-norm, p ∈ Q, p ≥ 1. We prove the NP-hardness of the problem and propose different mixed integer linear programming formulations. Moreover, we develop alternative Benders decomposition algorithms for the problem. Finally, we report some computational results comparing the formulations and the Benders decompositions on a set of instances

Application of decomposition techniques in a wildfire suppression optimization model

Resource assignment and scheduling models provides an automatic and fast decision support system for wildfire suppression logistics. However, this process generates challenging optimization problems in many real-world cases, and the computational time becomes a critical issue, especially in realistic-size instances. Thus, to overcome that limitation, this work studies and applies a set of decomposition techniques such as augmented Lagrangian, branch and price, and Benders decomposition’s to a wildfire suppression model. Moreover, a reformulation strategy, inspired by Benders’ decomposition, is also introduced and demonstrated. Finally, a numerical study comparing the behavior of the proposals using different problem sizes is conductedThis research work is supported by the R+D+I project grants PID2020-116587GB-I00 and PID2021-124030NB (C31 and C32), funded by MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”/EU. Second author investigation is funded by the Xunta de Galicia (contract post-doctoral 2019-2022). We acknowledge the computational resources provided by CESGA. Third author acknowledges support from the Xunta de Galicia through the ERDF (ED431C-2020-14 and ED431G 2019/01), and “CITIC”S

Spatial and temporal hierarchical decomposition methods for the optimal power flow problem

The subject of this thesis is the development of spatial and temporal decomposition methods for the optimal power flow problem, such as in the transmissiondistribution network topologies. In this context, we propose novel decomposition interfaces and effectivemethodology for both the spatial and temporal dimensions applicable to linear and non-linear representations of the OPF problem. These two decomposition strategies are combined with a Benders-based algorithmand have advantages in model building time, memory management and solving time. For example, in the 2880-period linear problems, the decomposition finds optimal solutions up to 50 times faster and allows even larger instances to be solved; and in multi-period non-linear problems with 48 periods, close-to-optimal feasible solutions are found 7 times faster. With these decompositions, detailed networks can be optimized in coordination, effectively exploiting the value of the time-linked elements in both transmission and distribution levels while speeding up the solution process, preserving privacy, and adding flexibility when dealing with different models at each level. In the non-linear methodology, significant challenges, such as active set determination, instability and non-convex overestimations, may hinder its effectiveness, and they are addressed, making the proposed methodology more robust and stable. A test network was constructed by combining standard publicly available networks resulting in nearly 1000 buses and lines with up to 8760 connected periods; several interfaces were presented depending on the problemtype and its topology using a modified Benders algorithm. Insight was given into why a Benders-based decomposition was used for this type of problem instead of a common alternative: ADMM. The methodology is useful mainly in two sets of applications: when highly detailed long-termlinear operational problems need to be solved, such as in planning frameworks where the operational problems solved assume no prior knowledge; and in full AC-OPF problems where prior information from historic solutions can be used to speed up convergence

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Decomposition methods for large-scale network expansion problems

Network expansion problems are a special class of multi-period network design problems in which arcs can be opened gradually in different time periods but can never be closed. Motivated by practical applications, we focus on cases where demand between origin-destination pairs expands over a discrete time horizon. Arc opening decisions are taken in every period, and once an arc is opened it can be used throughout the remaining horizon to route several commodities. Our model captures a key timing trade-off: the earlier an arc is opened, the more periods it can be used for, but its fixed cost is higher, since it accounts not only for construction but also for maintenance over the remaining horizon. An overview of practical applications indicates that this trade-off is relevant in various settings. For the capacitated variant, we develop an arc-based Lagrange relaxation, combined with local improvement heuristics. For uncapacitated problems, we develop four Benders decompositi

Modeling Price Formation in a Multi-Commodity Market - A Graph-Theoretical Decomposition Approach to Complexity Reduction

This thesis presents an optimization model to simulate the global price formation of multiple commodities over multiple time periods. The model considers the connection of commodities through their production processes. The supply side maximizes its total profit taking account of the price-demand relationships of all products. The variables of this model are production quantities, transport quantities, storage quantities, and commodity prices. We apply the model to a part of the petrochemical market. A large multi-commodity model requires many parameters. Moreover, the interpretation of the simulation results can become difficult. Therefore, this thesis focuses on the model and complexity reduction with respect to optimization models. We propose a graph-theoretical approach to reveal the structure of large block-separable problems and to compare different decompositions into subproblems. The connections between primal and dual variables of a constrained optimization problem are represented on a hypergraph, which can be analyzed and beneficially partitioned using appropriate graph-theoretical methods. We show how different partitions of the hypergraph constitute different decompositions of the optimization problem. Furthermore, we address the approximation of subproblems. The decomposition approach is adapted to the commodity market model. We formulate the subproblems for chosen sets of products and processes and present an algorithm for the automated identification of model components that are suited for an aggregation. The aggregation of components of the market model in terms of approximating subproblems is discussed from different points of view. Furthermore, we conduct sensitivity analyses within the overall problem and within subproblems. The numerical results of the application to a petrochemical market model reveal different possibilities of model reduction