Boundary Decomposition for Finding Nadir Objective Vector in Multi-Objective Discrete Optimization

The exact nadir objective vector of a multi-objective discrete optimization problem (MODOP) is crucial for decision-making but remains challenging to find. Existing methods for tackling this issue have limitations in theoretical guarantees or high computational costs. This paper applies boundary decomposition to the MODOP and proposes an exact algorithm called BDNC. BDNC is designed to address a bilevel optimization problem for each objective with finite-time convergence guarantees. The lower-level optimization problem, termed the boundary subproblem, is a scalarization of the MODOP. It can be solved using any suitable single-objective exact solver. According to the theoretical foundations of boundary decomposition, some specific settings of the boundary subproblem can ensure alignment with the nadir objective vector under mild conditions. The upper-level optimization problem evaluates a potential setting using the optimal solution to the lower-level one. It employs our proposed novel pruning method to efficiently identify the specific settings. Moreover, BDNC can leverage a trade-off provided by the decision-makers, potentially facilitating the decision-making process. Experiments on various MODOPs demonstrate that BDNC exhibits superior and reliable performance in terms of runtime compared to existing exact methods.

Authors

• Ruihao Zheng Southern University of Science and Technology
• Zhenkun Wang Southern University of Science and Technology

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