Improving logic-based Benders’ algorithms for solving min-max regret problems

• Published in: Badania Operacyjne i Decyzje/Operations Research and Decisions Vol. 31; no. 2; pp. 23 - 57
• Main Authors: Assunção, Lucas, Santos, Andréa Cynthia, Noronha, Thiago F., Andrade, Rafael
• Format: Journal Article
• Language: English
• Published: Wroclaw University of Science and Technology 2021; Wrocław University of Science and Technology
• Subjects: Computer Science; Operations Research
• ISSN: 2081-8858; 2391-6060; 1230-1868; 2391-6060
• DOI: 10.37190/ord210202

This paper addresses a class of problems under interval data uncertainty, composed of min-max regret generalisations of classical 0-1 optimisation problems with interval costs. These problems are called robust-hard when their classical counterparts are already NP-hard. The state-of-the-art exact algorithms for interval 0-1 min-max regret problems in general work by solving a corresponding mixed integer linear programming formulation in a Benders’ decomposition fashion. Each of the possibly exponentially many Benders’ cuts is separated on the fly through the resolution of an instance of the classical 0-1 optimisation problem counterpart. Since these separation subproblems may be NP-hard, not all of them can be easily modelled by means of Linear Programming (LP), unless P = NP. In this work, we formally describe these algorithms through a logic-based Benders’ decomposition framework and assess the impact of three warm-start procedures. These procedures work by providing promising initial cuts and primal bounds through the resolution of a linearly relaxed model and an LP-based heuristic. Extensive computational experiments in solving two challenging robust-hard problems indicate that these procedures can highly improve the quality of the bounds obtained by the Benders’ framework within a limited execution time. Moreover, the simplicity and effectiveness of these speed-up procedures makes them an easily reproducible option when dealing with interval 0-1 min-max regret problems in general, especially the more challenging subclass of robust-hard problems