We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliar variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and Strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.

翻译：本文研究了一类双层成像学习问题，其中下层实例对应于涉及一阶和二阶非光滑稀疏正则化项的凸变分模型。通过利用下层问题原始-对偶重构的几何性质并引入合适的辅助变量，我们成功将原始双层问题转化为带互补约束的数学规划（MPCC）。针对转化后的MPCC，我们证明了严格的约束规范条件（MPCC-RCPLD和部分MPCC-LICQ），并推导了Mordukhovich（M-）和强（S-）平稳性条件。MPCC的平稳性系统同时也转化为原始问题的平稳性条件。此外，我们推导了二阶充分最优性条件，并得到了平稳点的局部唯一性结果。所提出的重构方法可推广至函数空间中的问题，从而得到带有状态梯度约束的MPCC。MPCC重构还促进了现有大规模非线性规划求解器的高效应用，正如在针对不同成像应用研究的姊妹论文中所展示的那样。