Time-discrete Consensus-Based Optimization Algorithm for Multi-Objective Optimization Problems

In this paper, we study asymptotic consensus and finite-time reachability to Pareto optimal set for time-discrete multi-objective consensus-based optimization (M-CBO) algorithm in particle regime without resorting to the mean-field approximation. While the original CBO algorithm [J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci. 28 (2018) 1037–1066] 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 [G. Borghi, M. Herty and L. Pareschi, An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation, Appl. Math. Optim. 88 (2023) 58].