  A novel multiscale consensus-based optimization (CBO) algorithm for solving
bi- and tri-level optimization problems is introduced. Existing CBO techniques
are generalized by the proposed method through the employment of multiple
interacting populations of particles, each of which is used to optimize one
level of the problem. These particle populations are evolved through
multiscale-in-time dynamics, which are formulated as a singularly perturbed
system of stochastic differential equations. Theoretical convergence analysis
for the multiscale CBO model to an averaged effective dynamics as the
time-scale separation parameter approaches zero is provided. The resulting
algorithm is presented for both bi-level and tri-level optimization problems.
The effectiveness of the approach in tackling complex multi-level optimization
tasks is demonstrated through numerical experiments on various benchmark
functions. Additionally, it is shown that the proposed method performs well on
min-max optimization problems, comparing favorably with existing CBO algorithms
for saddle point problems.
