Global optimization of a non-convex objective function often appears in large-scale machine learning and artificial intelligence applications. Recently, consensus-based optimization (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 [J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, https://arxiv.org/abs/1909.09249v1]. Prior to this 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 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 [J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, https://arxiv.org/abs/1909.09249v1] 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.

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We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.