  We present stochastic consensus and convergence of the discrete
consensus-based optimization (CBO) algorithm with random batch interactions and
heterogeneous external noises. Despite the wide applications and successful
performance in many practical simulations, the convergence of the discrete CBO
algorithm was not rigorously investigated in such a generality. In this work,
we introduce a generalized discrete CBO algorithm with a weighted
representative point and random batch interactions, and show that the proposed
discrete CBO algorithm exhibits stochastic consensus and convergence toward the
common equilibrium state exponentially fast under suitable assumptions on
system parameters. For this, we recast the given CBO algorithm with random
batch interactions as a discrete consensus model with a random switching
network topology, and then we use the mixing property of interactions over
sufficiently long time interval to derive stochastic consensus and convergence
estimates in mean square and almost sure senses. Our proposed analysis
significantly improves earlier works on the convergence analysis of CBO models
with full batch interactions and homogeneous external noises.
