用于连续和离散时间结构化双层优化的正则化外梯度方法

中文摘要 AI 辅助

在实希尔伯特空间中，我们研究一类双层优化问题，该问题旨在最小化外层凸函数，使其取值在极大单调算子的零点集上。在光滑场景下，外层目标函数为凸且Fréchet可微，内层算子为单值、连续且单调，我们为该问题关联了一个一阶动力系统，可视为应用于动态正则化算子的单调流。在内层问题满足合适的几何条件——要么是弱Attouch-Czarnecki型可积性条件，要么是更强的尖锐性假设——的情况下，我们建立了外层残差和内层残差的最后迭代收敛速率，同时证明了轨迹弱收敛至该双层问题的最优解。在光滑+非光滑场景下，我们为外层目标函数添加了一个真凸下半连续函数，同时为内层算子添加了一个具有相同性质的函数的次微分。我们提出了一种正则化近端外梯度算法，其中前向和后向步骤分别针对动态正则化算子和函数执行。在内层问题满足与光滑场景类似的几何假设的情况下，我们建立了外层残差和内层残差的最后迭代收敛速率，同时证明了迭代序列弱收敛至该双层问题的最优解。

英文摘要

In a real Hilbert space, we study a bilevel optimization problem that consists in minimizing an outer convex function over the zero set of a maximally monotone operator. In the smooth setting, where the outer objective is convex and Fréchet differentiable and the inner operator is single-valued, continuous and monotone, we associate with the problem a first-order dynamical system that can be viewed as a monotone flow applied to a dynamically regularized operator. Under a suitable Attouch-Czarnecki-type condition for the inner level, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the trajectories to optimal solutions of the bilevel problem. In the smooth+nonsmooth setting, we enrich the outer objective with a proper, convex, and lower semicontinuous function, while the inner operator is augmented by the subdifferential of a function with the same properties. We propose a regularized proximal-extragradient algorithm in which both the forward and backward steps are performed with respect to dynamically regularized operators and functions, respectively. Under an analogous Attouch-Czarnecki-type condition formulated for the smooth+nonsmooth case, we establish last-iterate convergence rates for both the outer and inner residuals, together with weak convergence of the iterates to optimal solutions of the bilevel problem.

发表机构

• University of Vienna（维也纳大学）

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