In this work we propose MirrorCBO, a consensus-based optimization (CBO) method which generalizes standard CBO in the same way that mirror descent generalizes gradient descent. For this we apply the CBO methodology to a swarm of dual particles and retain the primal particle positions by applying the inverse of the mirror map, which we parametrize as the subdifferential of a strongly convex function $\phi$. In this way, we combine the advantages of a derivative-free non-convex optimization algorithm with those of mirror descent. As a special case, the method extends CBO to optimization problems with convex constraints. Assuming bounds on the Bregman distance associated to $\phi$, we provide asymptotic convergence results for MirrorCBO with explicit exponential rate. Another key contribution is an exploratory numerical study of this new algorithm across different application settings, focusing on (i) sparsity-inducing optimization, and (ii) constrained optimization, demonstrating the competitive performance of MirrorCBO. We observe empirically that the method can also be used for optimization on (non-convex) submanifolds of Euclidean space, can be adapted to mirrored versions of other recent CBO variants, and that it inherits from mirror descent the capability to select desirable minimizers, like sparse ones. We also include an overview of recent CBO approaches for constrained optimization and compare their performance to MirrorCBO.

翻译：本文提出MirrorCBO方法，这是一种共识优化方法，其与标准共识优化的关系正如镜像下降与梯度下降的关系。为此，我们将共识优化框架应用于对偶粒子群，并通过应用镜像映射的逆映射（参数化为强凸函数$\phi$的次微分）来保留原始粒子位置。通过这种方式，我们结合了无导数非凸优化算法与镜像下降方法的优势。作为特例，该方法将共识优化扩展至具有凸约束的优化问题。在假设与$\phi$相关的Bregman距离有界的前提下，我们给出了MirrorCBO具有显式指数收敛速率的渐近收敛性证明。另一项关键贡献是通过不同应用场景的探索性数值研究，重点关注（i）稀疏诱导优化与（ii）约束优化问题，展示了MirrorCBO具有竞争力的性能。我们通过实验观察到，该方法还可用于欧几里得空间（非凸）子流形上的优化，能够适配其他近期共识优化变体的镜像版本，并且继承了镜像下降选择理想极小值（如稀疏解）的能力。本文还综述了近期用于约束优化的共识优化方法，并将其性能与MirrorCBO进行了对比。