Editorial

Ann Oper Res (2013) 210:1–4 DOI 10.1007/s10479-013-1490-5 Editorial Georgios K.D. Saharidis · Marianthi Ierapetritou · Christodoulos A. Floudas Published online: 19 October 2013 © Springer Science+Business Media New York 2013 In optimization, the decision maker is frequently not contented with the relatively long solution times resulting from state-of-the-art solvers in linear or mixed-integer linear models. On one hand, input and decision variables may grow fast in order to capture the nature of the underlying problem in its full scale; on the other hand, the time that the decision maker or the beneficiary is willing to wait for an optimal or even a good enough solution to the problem may be extremely limited. Frequently, there are cases in which an out-of-the-box solver is unable to provide a solution to the original problem in its large-scale standard form. Regardless of the cause, and even though optimality may often be a relatively mild requirement, the solution needs to be characterized and accompanied by its distance from the optimum. Exact methods need to be employed to that end. Decomposition techniques have shown that they can satisfactorily address the above concerns in considerably shorter computation times. In this context, Benders decomposition has attracted the significant interest of many researchers throughout the years and has been C.A. Floudas is Stephen C. Macaleer ’63 Professor in Engineering and Applied Science and Professor of Chemical and Biological Engineering. B Dr. G.K.D. Saharidis ( ) Department of Mechanical Engineering, University of Thessaly, Volos, 38334, Greece e-mail: url: http://www.mie.uth.gr/n_one_staff.asp?cid=1&id=243 Dr. M. Ierapetritou Department of Chemical and Biochemical Engineering, Rutgers—The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854-8058, USA e-mail: url: http://sol.rutgers.edu/staff/marianth/ Dr. C.A. Floudas Department of Chemical and Biological Engineering, Princeton University Princeton, NJ 08544-5263, USA e-mail: url: http://titan.princeton.edu 2 Ann Oper Res (2013) 210:1–4 a popular approach in tackling various types of optimization problems including, most recently, stochastic and bilevel optimization. This special volume is dedicated to the theory and applications of the Benders decomposition method and aims to showcase the benefits acquired when the method is correctly fine-tuned to match the peculiarities and specificities of each problem. In a Benders decomposition framework, a great number of decision parameters can dramatically influence the performance of the algorithms. For instance, the separation of the original problem into master and sub-problem is not a de facto choice and may be decided in several different ways. The reader will see that the way the authors define these two distinct components often reflects the inherent nature of the problem itself. The settings and communication protocols between the two may also be configured: the incumbent solution that will be communicated by the master to the sub-problem as well as the number or type of cuts that the sub-problem will contribute to the master, both constitute structural decisions in the solution strategy. A warm start given to the master problem may also boost solution time and can be beneficial depending on the nature of the problem addressed. The guest editors wish to thank the authors of this special volume for having contributed to clarifying some of the above as well as additional important points in their work; we are sure that their individual contributions will provide the readers with further insight into the power and efficiency of Benders decomposition. A brief description of this special volume’s contributions follows. Shim et al. present a branch-and-bound algorithm for discretely constrained mathematical programs with equilibrium constraints. The authors present a dynamic partition scheme to overcome the non-convexity of the Benders sub-problem which ensures convergence to the global optimum. Naoum-Sawaya and Elhedhli present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). They show that the ACCPM-based Benders cuts are both pareto-optimal and global, making them valid for any node of the branch-and-bound tree. The global cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multi-commodity capacitated fixed charge network design problem. Sherali and Lunday also explore certain algorithmic strategies for accelerating the convergence of the Benders decomposition method via the generation of maximal non-dominated cuts. The authors propose an algorithmic strategy that utilizes a preemptively small perturbation of the right-hand side of the Benders sub-problem to generate maximal non-dominated Benders cuts, as well as a complementary strategy that generates an additional cut at each iteration. Lazimy proposes an interactive polyhedral outer approximation method to solve a broad class of multi-objective optimization problems which may include nonlinear and non-differentiable objective and constraint functions, and continuous or discrete decision variables. The classical implementation of Benders decomposition in some cases results in low density Benders cuts. Covering Cut Bundle (CCB) generation addresses this issue in a novel way, generating a bundle of cuts that could cover more decision variables of the Benders master problem than the classical Benders cut. The motivation to improve further CCB generation led to three cut generation strategies (Saharidis and Ierapetritou, Nader et al., and Tang et al.). These strategies are referred to as the Maximum Density Cut (MDC) generation strategies. MDC are based on the observation that in some cases, when applying CCB generation, it is computationally expensive to cover the maximum number of master decision variables. MDC strategies address this issue by generating the cut that involves the rest of the decision variables of the master problem that are not covered in the Benders cut and/or Ann Oper Res (2013) 210:1–4 3 in the CCB. An extension of MDC strategies was presented by Tang et al. who investigate a logistics facility location problem to determine whether the existing facilities remain open or not, what the expansion size of the open facilities should be, and which potential facilities should be selected. The authors propose three groups of valid inequalities and a high density Pareto cut generation method to accelerate the convergence by lifting Pareto-optimal cuts. MDC strategies and their extensions can be applied as a complementary step to the CCB generation or as a standalone strategy. Finally, You and Grossmann present a multicut version of the Benders decomposition method for s (...truncated)