We present a rigorous convergence analysis of a new method for density-based topology optimization: Sigmoidal Mirror descent with a Projected Latent variable. SiMPL provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in a companion article. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. Regardless of the line search algorithm, SiMPL delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm and superior performance over the two most popular first-order methods in topology optimization: OC and MMA.

翻译：本文对一种新型密度基拓扑优化方法——带投影潜变量的S型镜像下降法（SiMPL）进行了严格的收敛性分析。SiMPL方法能够提供逐点有界保持的设计更新，并比其他常用的一阶拓扑优化方法具有更快的收敛速度。得益于其强大的边界保持特性，该方法展现出卓越的鲁棒性，这一点在本文及姊妹篇的众多算例中均得到验证。此外，该方法结构清晰、更新表达式解析，易于实现。我们的分析涵盖该方法的两个版本，其区别在于采用的线搜索策略：改进的Armijo回溯线搜索和Bregman回溯线搜索。无论采用何种线搜索算法，SiMPL都能保证目标函数的严格单调递减，并具有更直观的收敛特性，例如密度变量在活跃集上的强逐点收敛性、增量范数收敛至零等。数值实验进一步表明，该算法具有明显的网格无关收敛特性，其性能优于拓扑优化领域两种最常用的一阶方法：OC（优化准则法）和MMA（移动渐近线法）。