Acceleration and Global Convergence of a First-Order Primal-Dual Method for Nonconvex Problems
研究了修正原始对偶混合梯度法在非线性算子导致的非凸问题中的扩展，提出了新的步长参数条件以实现加速和全局收敛，并在PDE约束优化中验证了有效性。
The primal-dual hybrid gradient method, modified (PDHGM, also known as the Chambolle--Pock method), has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step-length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step-length rules for PDE-constrained optimization problems.