With a view on bilevel and PDE-constrained optimisation, we develop iterative estimates $\widetilde{F'}(x^k)$ of $F'(x^k)$ for compositions $F :=J \circ S$, where $S$ is the solution mapping of the inner optimisation problem or PDE. The idea is to form a single-loop method by interweaving updates of the iterate $x^k$ by an outer optimisation method, with updates of the estimate by single steps of standard optimisation methods and linear system solvers. When the inner methods satisfy simple tracking inequalities, the differential estimates can almost directly be employed in standard convergence proofs for general forward-backward type methods. We adapt those proofs to a general inexact setting in normed spaces, that, besides our differential estimates, also covers mismatched adjoints and unreachable optimality conditions in measure spaces. As a side product of these efforts, we provide improved convergence results for nonconvex Primal-Dual Proximal Splitting (PDPS).

翻译：针对双层优化和偏微分方程约束优化问题，我们针对复合函数$F :=J \circ S$（其中$S$为内层优化问题或偏微分方程的解映射）开发了$F'(x^k)$的迭代估计$\widetilde{F'}(x^k)$。该方法通过将外层优化方法对迭代点$x^k$的更新，与标准优化方法和线性系统求解器对估计值的单步更新交织进行，从而形成单循环算法。当内层方法满足简单跟踪不等式时，差分估计几乎可直接应用于一般前向-后向型方法的标准收敛性证明。我们将这些证明推广到赋范空间中的一般非精确设定，该设定不仅涵盖我们的差分估计，还包含测度空间中的失配伴随算子和不可达最优性条件。作为这些工作的副产品，我们为非凸原始-对偶近端分裂算法提供了改进的收敛性结果。