SIAM Journal on Imaging Sciences, Volume 16, Issue 3, Page 1655-1686, September 2023.
Abstract. 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 auxiliary 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 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.

中文翻译：

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作为具有互补约束的数学程序的双层成像学习问题：重新表述和理论

SIAM 影像科学杂志，第 16 卷，第 3 期，第 1655-1686 页，2023 年 9 月。
抽象的。我们研究了一系列双层成像学习问题，其中较低级别的实例对应于涉及一阶和二阶非平滑稀疏性正则化器的凸变分模型。通过利用低层问题的原对偶重新表述的几何性质并引入合适的辅助变量，我们能够将原始双层问题重新表述为具有互补约束的数学程序（MPCC）。对于后者，我们证明了严格约束条件（MPCC-RCPLD 和部分 MPCC-LICQ），并推导出 Mordukhovich (M-) 和强 (S-) 平稳性条件。MPCC 的平稳性系统也转变为原始公式的平稳性条件。还导出了二阶充分最优性条件，以及驻点的局部唯一性结果。所提出的重构可以扩展到函数空间中的问题，从而导致 MPCC 对状态梯度有约束。MPCC 重构还可以有效利用可用的大规模非线性编程求解器，如研究不同成像应用的配套论文中所示。