  We propose Discrete Consensus-Based Optimization (DCBO), a fully discrete
version of the Consensus-Based Optimization (CBO) framework. DCBO is a
multi-agent method for the global optimization of possibly non-convex and
non-differentiable functions. It aligns with the CBO paradigm, which promotes a
consensus among agents towards a global optimum through simple stochastic
dynamics amenable to rigorous mathematical analysis. Despite the promises,
there has been a gap between the analysis of CBO and the actual behavior of the
agents from its time-discrete implementation, as the former has focused on the
system of continuous stochastic differential equations defining the model or
its mean-field approximation. In particular, direct analysis of CBO-type
algorithms with heterogeneous stochasticity is very challenging. DCBO
distinguishes itself from these approaches in the sense that it has no
continuous counterpart, thanks to the replacement of the "softmin" operator
with the "hardmin" one, which is inherently discrete. Yet, it maintains the
operational principles of CBO and allows for rich mathematical analysis. We
present conditions, independent of the number of agents, for achieving a
consensus or convergence and study the circumstances under which global
optimization occurs. We test DCBO on a large number of benchmark functions to
show its merits. We also demonstrate that DCBO is applicable to a diverse range
of real-world problems, including neural network training, compressed sensing,
and portfolio optimization, with competitive performance.
