We derive an extension of the sequential homotopy method that allows for the application of inexact Krylov methods for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.

翻译：我们推导了序列同伦法的一种扩展形式，允许对同伦子问题的局部半光滑牛顿法中出现的线性（双）鞍点系统应用不精确Krylov方法。针对一类在拉格朗日函数海森矩阵的非对角块中（经过适当的变量划分后）存在零元素的问题，我们提出并分析了一种基于双舒尔补方法的高效、可并行化的对称正定预处理器。对于具有PDE约束的离散最优控制问题，这种结构通常通过将变量规范划分为状态变量和控制变量而呈现。最后，我们给出一个含椭圆偏微分方程与控制约束的病态高度非线性基准优化问题的数值结果。所提出的方法在二维基准问题上的求解速度快于直接线性代数方法，并支持大规模三维问题的并行求解。