We derive an extension of the sequential homotopy method that allows for the application of inexact solvers 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 allows for the parallel solution of large 3D problems.

翻译：我们推导了序贯同伦法的一种扩展形式，使得能够对同伦子问题的局部半光滑牛顿法中出现的线性（双）鞍点系统应用非精确求解器。针对拉格朗日函数Hessian矩阵的非对角块存在零元素的问题类别（经过适当的变量划分后），我们提出并分析了一种基于双Schur补方法的高效、可并行化的对称正定预条件子。对于具有偏微分方程约束的离散最优控制问题，这种结构通常出现在将变量按状态和控制进行规范划分的情形中。最后，我们给出了一个带有椭圆型偏微分方程和控制约束的、条件数恶劣且高度非线性的基准优化问题的数值结果。所提方法支持大规模三维问题的并行求解。