Boundary Decomposition for Nadir Objective Vector Estimation

The nadir objective vector plays a key role in solving multi-objective optimization problems (MOPs), where it is often used to normalize the objective space and guide the search. The current methods for estimating the nadir objective vector perform effectively only on specific MOPs. This paper reveals the limitations of these methods: exact methods can only work on discrete MOPs, while heuristic methods cannot deal with the MOP with a complicated feasible objective region. To fill this gap, we propose a general and rigorous method, namely boundary decomposition for nadir objective vector estimation (BDNE). BDNE scalarizes the MOP into a set of boundary subproblems. By utilizing bilevel optimization, boundary subproblems are optimized and adjusted alternately, thereby refining their optimal solutions to align with the nadir objective vector. We prove that the bilevel optimization identifies the nadir objective vector under mild conditions. We compare BDNE with existing methods on various black-box MOPs. The results conform to the theoretical analysis and show the significant potential of BDNE for real-world application.