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arXiv · 2504.10990

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

Jinhuan Wang·Keyu Li·Hui Huang
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Abstract

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 $L^\infty(0,T;L^1\cap L^\infty(\mathbb{R}^d))$. Furthermore, we show that the weak solutions exhibit improved $H^2$-regularity when the initial data is regular.

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Jinhuan Wang, Keyu Li, Hui Huang. 2025-04-15. Mathematical Analysis of the PDE Model for the Consensus-based Optimization. https://arxiv.org/abs/2504.10990

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