It is well known that mirror descent may diverge or cycle on merely monotone variational inequalities. In this paper, we propose \emph{Target Mirror Descent} (TMD), a unified framework that stabilizes monotone flows via a target point correction mechanism in the dual update. By appropriate design choices, TMD recovers the proximal point algorithm, extragradient methods, splitting methods, Brown-von Neumann-Nash dynamics, forward-backward-forward dynamics, and discounted mirror descent as special cases. Thus, we establish a unified perspective on these landmark algorithms and their convergence. Beyond unification, we leverage the TMD framework to correct an equilibrium misalignment in discounted mirror descent and to generalize its higher-order extension beyond interior solutions. Moreover, a key structural feature of TMD is the explicit decoupling of the mirror map from the target determination, which enables \emph{geometric ensembles}: multiple algorithms solve the same problem in parallel using distinct mirror maps, while sharing a common dual update. We show that such an ensemble rigorously reduces to a single TMD with a synthesized mirror map, and thus inherits these convergence guarantees.

翻译：众所周知，镜像下降法在仅满足单调性的变分不等式上可能发散或产生循环。本文提出目标镜像下降（TMD）——一种通过对偶更新中的目标点校正机制来稳定单调流的统一框架。通过适当的设计选择，TMD可以恢复近端点算法、外梯度方法、分裂方法、Brown-von Neumann-Nash动力学、前向-后向-前向动力学以及折扣镜像下降作为特例。由此，我们建立了这些经典算法及其收敛性的统一视角。超越统一性之外，我们利用TMD框架纠正了折扣镜像下降中的均衡错位问题，并将其高阶推广扩展到内点解之外。此外，TMD的一个关键结构特征是镜像映射与目标确定的显式解耦，这实现了几何集成：多种算法并行解决同一问题，各自使用不同的镜像映射，同时共享一个共同的对偶更新。我们证明，这种集成严格等价于一个具有合成镜像映射的单一TMD，因此继承了其收敛保证。