We study asymptotic consensus and finite-time reachability to Pareto optimal set for time-discrete multiobjective consensus-based optimization (M-CBO) algorithm in particle regime without resorting to the meanfield approximation. While the original CBO algorithm 15 assumes that all particles share common single objective function, the M-CBO algorithm assigns each particle to a distinct sub-objective function obtained by the method of scalarization. In this paper, we show that for a sufficiently small noise, the expected value of the state diameter decays to zero exponentially fast. This results in almost sure asymptotic consensus. We also show that the CBO particle system approximates a Pareto optimal set in finite time. Moreover, the approximation becomes increasingly accurate, as the noise strength vanishes, while the number of particles and the inverse temperature parameter tend to infinity. These results extend our theoretical understanding for CBO-type algorithms to the multi-objective regime by exhibiting asymptotic behaviors that are not observed in the previous mean-field analysis for M-CBO in 11 .
Kang et al. (Tue,) studied this question.