Invited ReviewThe Benders decomposition algorithm: A literature review
Introduction
It has been more than five decades since the Benders Decomposition (BD) algorithm was proposed by Benders (1962), with the main objective of tackling problems with complicating variables, which, when temporarily fixed, yield a problem significantly easier to handle. The BD method (also referred to as variable partitioning, Zaourar and Malick (2014), and outer linearization, Trukhanov, Ntaimo, and Schaefer (2010)) has become one of the most widely used exact algorithms, because it exploits the structure of the problem and decentralizes the overall computational burden. Successful applications are found in many divers fields, including planning and scheduling (Canto, 2008, Hooker, 2007), health care (Luong, 2015), transportation and telecommunications (Costa, 2005), energy and resource management (Cai, McKinney, Lasdon, Watkins, 2001, Zhang, Ponnambalam, 2006), and chemical process design (Zhu & Kuno, 2003), as illustrated in Table 1.
The BD method is based on a sequence of projection, outer linearization, and relaxation (Geoffrion, 1970a, Geoffrion, 1970b). The model is first projected onto the subspace defined by the set of complicating variables. The resulting formulation is then dualized, and the associated extreme rays and points respectively define the feasibility requirements (feasibility cuts) and the projected costs (optimality cuts) of the complicating variables. Thus, an equivalent formulation can be built by enumerating all the extreme points and rays. However, performing this enumeration and, then, solving the resulting formulation is generally computationally exhausting, if not impossible. Hence, one solves the equivalent model by applying a relaxation strategy to the feasibility and optimality cuts, yielding a Master Problem (MP) and a subproblem, which are iteratively solved to respectively guide the search process and generate the violated cuts.
The BD algorithm was initially proposed for a class of mixed-integer linear programming (MILP) problems. When the integer variables are fixed, the resulting problem is a continuous linear program (LP) for which we can use standard duality theory to develop cuts. Many extensions have since been developed to apply the algorithm to a broader range of problems (e.g., Geoffrion, 1972, Hooker, Ottosson, 2003). Other developments were proposed to increase the algorithm’s efficiency on certain optimization classes (e.g., Costa, Cordeau, Gendron, Laporte, 2012, Crainic, Hewitt, Rei, 2014). In addition, BD often provides a basis for the design of effective heuristics for problems that would otherwise be intractable (Côté, Laughton, 1984, Raidl, 2015). The BD approach has thus become widely used for linear, nonlinear, integer, stochastic, multi-stage, bilevel, and other optimization problems, as illustrated in Table 2.
Fig. 1 depicts the increasing interest in the BD algorithm over the years. Despite this level of interest, there has been no comprehensive survey of the method in terms of its numerical and theoretical challenges and opportunities. The now out-of-date survey by Costa (2005) reviews only applications to fixed-charge network design problems. The main goal of this paper therefore is to contribute to filling this gap by reviewing the current state-of-the-art, focusing on the main ideas for accelerating the method, discussing the main variants and extensions aimed to handle more general problems involving, e.g., nonlinear/integer/constraint programming subproblems, and identifying trends and promising research directions. Many different enhancement strategies were proposed to address the shortcomings of the BD method and accelerate it. This effort contributed significantly to the success of the method. We propose a taxonomy of the enhancement and acceleration strategies based on the main components of the algorithm: the decomposition strategy, the strategies to handle the MP and subproblem, and the strategies to generate solutions and cuts. The taxonomy provides the framework to classify and synthesize the literature and to identify relations among strategies and between these and the BD method.
The remainder of this paper is organized as follows. Section 2 presents the classical BD algorithm, the associated model selection criteria, and its relationship to other decomposition methods. Section 3 presents the proposed taxonomy, used to survey the acceleration strategies in Sections 4–7. Section 8 presents Benders-type heuristics, and Section 9 describes extensions of the classical algorithm. Finally, Section 10 provides concluding remarks and describes promising research directions.
Section snippets
The Benders decomposition method
We present in this section the classical version of the Benders algorithm (Benders, 1962). We review its extensions to a broader range of optimization problems in Section 9.
Taxonomy of the enhancement strategies
A straightforward application of the classical BD algorithm may require excessive computing time and memory (Magnanti, Wong, 1981, Naoum-Sawaya, Elhedhli, 2013). Its main drawbacks include: time-consuming iterations; poor feasibility and optimality cuts; ineffective initial iterations; zigzagging behavior of the primal solutions and slow convergence at the end of the algorithm (i.e., a tailing-off effect); and upper bounds that remain unchanged in successive iterations because equivalent
Decomposition strategies
Recent studies have presented various modified decomposition strategies. Crainic, Hewitt, Rei, 2014, Crainic, Hewitt, Rei, 2016 emphasized that the BD method causes the MP to lose all the information associated with the noncomplicating variables. This results in instability, erratic progression of the bounds, and a large number of iterations. Moreover, the problem structure associated with the linking constraints (3) is lost, and thus many classical VIs are not applicable. The authors proposed
Solution procedure
The iterative solution of the MP and subproblem is a major computational bottleneck. In particular, the MP, an MILP formulation, is often lacking special structure, and is continually growing in size becoming more and more difficult to solve. Classically, the MP is solved to optimality via branch-and-bound, while the subproblem is handled with the simplex method. In this section, we survey the various alternatives that have been proposed. These strategies exploit the structure of the MP and
Solution generation
The quality of the solutions for the set of complicating variables directly determines the number of iterations, as they are used to generate cuts and bounds. These solutions are traditionally found by exactly or approximately solving the regular MP. Three approaches have been proposed to improve the quality of the solutions or generate them more quickly: (1) using alternative formulations, (2) improving the MP formulation, and (3) using heuristics to independently generate solutions or to
Cut generation
The number of iterations is closely related to the strength of the cuts, i.e., the values selected for the dual variables. Researchers have explored ways to select or strengthen the traditional feasibility and optimality cuts or to generate additional valid cuts.
Magnanti and Simpson (1978) and Magnanti and Wong (1981) were the first to consider the degeneracy of the subproblems, when the dual subproblem has multiple optimal solutions that do not yield cuts of equal strength. Hence, to find the
Benders-type heuristics
Because of time and memory limitations, the execution of the BD method might be stopped before its convergence is established. Moreover, in many practical applications, decision-makers do not need a provably optimal solution, a good feasible solution being deemed sufficient. Such a solution is often obtained somewhat early in the solution process.
From a heuristic point of view, the BD method is an attractive methodology because it can take advantage of special structures and provides a rich
Extensions of the classical Benders decomposition method
The classical BD algorithm was proposed for certain classes of MILPs for which the integer variables were considered to be complicating, and standard duality theory could be applied to the subproblem to develop cuts. Extensions of the method have allowed it to address a broader range of optimization problems, including integer subproblems (e.g., Carøe & Tind, 1998), nonlinear functions (e.g., Cai, McKinney, Lasdon, Watkins, 2001, Geoffrion, 1972), logical expressions (e.g., Eremin & Wallace,
Conclusions and future research
We have presented a state-of-the-art survey of the BD method. We have discussed the classical algorithm, the impact of the problem formulation on its convergence, and the relationship to other decomposition methods. We have developed a taxonomy to classify the literature on acceleration strategies, based on the main components of the algorithm, which provided rich guidelines to analyze various enhancements, identifying shortcomings, trends and potential research directions. We have also
Acknowledgments
Partial funding for this project has been provided by the Natural Sciences and Engineering Council of Canada (NSERC), through its Discovery Grant program and by the Fonds de recherche du Québec through its Team Grant program. We also gratefully acknowledge the support of Fonds de recherche du Québec through their strategic infrastructure grants.
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