Mathematical Analysis of the PDE Model for the Consensus-based Optimization

In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a diffusion term that is both singular and degenerate. By employing a regularization procedure in combination with a compactness argument, we establish the global existence and uniqueness of weak solutions in $L^{\mathrm{\infty }}(0,T;L^1\cap L^{\mathrm{\infty }}({\mathbb{R}}^d))$. Furthermore, we show that the weak solutions exhibit improved $H^2$-regularity when the initial data is regular.