  Global optimization of a non-convex objective function often appears in
large-scale machine-learning and artificial intelligence applications.
Recently, consensus-based optimization (in short CBO) methods have been
introduced as one of the gradient-free optimization methods. In this paper, we
provide a convergence analysis for the first-order CBO method in
\cite{C-J-L-Z}. Prior to the current work, the convergence study was carried
out for CBO methods on corresponding mean-field limit, a Fokker-Planck
equation, which does not imply the convergence of the CBO method {\it per se}.
Based on the consensus estimate directly on the first-order CBO model, we
provide a convergence analysis of the first-order CBO method \cite{C-J-L-Z}
without resorting to the corresponding mean-field model. Our convergence
analysis consists of two steps. In the first step, we show that the CBO model
exhibits a global consensus time asymptotically for any initial data, and in
the second step, we provide a sufficient condition on system parameters--which
is dimension independent-- and initial data which guarantee that the converged
consensus state lies in a small neighborhood of the global minimum almost
surely.
