Abstract
A general enhancement of the Benders’ decomposition (BD) algorithm can be achieved through the improved use of large neighbourhood search heuristics within mixed-integer programming solvers. While mixed-integer programming solvers are endowed with an array of large neighbourhood search heuristics, few, if any, have been designed for BD. Further, typically the use of large neighbourhood search heuristics is limited to finding solutions to the BD master problem. Given the lack of general frameworks for BD, only ad hoc approaches have been developed to enhance the ability of BD to find high quality primal feasible solutions through the use of large neighbourhood search heuristics. The general BD framework of SCIP has been extended with a trust region based heuristic and a general enhancement for large neighbourhood search heuristics. The general enhancement employs BD to solve the auxiliary problems of all large neighbourhood search heuristics to improve the quality of the identified solutions. The computational results demonstrate that the trust region heuristic and a general large neighbourhood search enhancement technique accelerate the improvement in the primal bound when applying BD.
Highlights
Benders’ decomposition (BD) (Benders 1962) is a popular mathematical programming technique that is used to solve large-scale optimisation problems
Considering this algorithmic design for Large neighbourhood search (LNS) heuristics, a factor contributing to the decrease in the primal integral by LNS check with respect to Benders is that the minimum number of nodes processed by Benders for the stochastic capacitated facility location problem (SCFLP) instances is 4094 and the shmean of nodes processed is 11,084
This paper presents two different, but complementary, approaches for improving the heuristic performance of the BD algorithm—the Large neighbourhood Benders’ search (LNBS) and the trust region (TR) heuristic
Summary
Benders’ decomposition (BD) (Benders 1962) is a popular mathematical programming technique that is used to solve large-scale optimisation problems. While BD is an effective exact solution algorithm for large-scale optimisation problems, there are many situations where finding high-quality feasible solutions can be challenging In such situations, BD is a valuable technique to exploit problem separability in the design of effective primal heuristic methods. The underlying mixed integer programming (MIP) solver and the included heuristics are critical to the ability of the BD algorithm in finding high-quality feasible solutions. This is important when implementing BD using the branch-and-cut approach, i.e. using callback functions available within a MIP solver, where more interaction with the solver is supported.
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