Backtracking linesearch is the de facto approach for minimizing continuously differentiable functions with locally Lipschitz gradient. In recent years, it has been shown that in the convex setting it is possible to avoid linesearch altogether, and to allow the stepsize to adapt based on a local smoothness estimate without any backtracks or evaluations of the function value. In this work we propose an adaptive proximal gradient method, adaPG, that uses novel estimates of the local smoothness modulus which leads to less conservative stepsize updates and that can additionally cope with nonsmooth terms. This idea is extended to the primal-dual setting where an adaptive three-term primal-dual algorithm, adaPD, is proposed which can be viewed as an extension of the PDHG method. Moreover, in this setting the "essentially" fully adaptive variant adaPD$^+$ is proposed that avoids evaluating the linear operator norm by invoking a backtracking procedure, that, remarkably, does not require extra gradient evaluations. Numerical simulations demonstrate the effectiveness of the proposed algorithms compared to the state of the art.

翻译：回溯线性搜索是处理具有局部利普希茨连续梯度的连续可微函数最小化问题的事实标准方法。近年来研究表明，在凸优化场景中，完全避免线性搜索是可行的——无需任何回溯或函数值评估，即可基于局部光滑度估计自适应调整步长。本文提出一种自适应近端梯度方法adaPG，其采用新颖的局部光滑模量估计技术，既能实现更保守的步长更新策略，又能处理非光滑项。该思想被进一步拓展至原始-对偶框架，提出自适应三项原始-对偶算法adaPD，可视为PDHG方法的推广。在此基础上，本文进一步提出"实质上"完全自适应变体adaPD$^+$，该算法通过引入无需额外梯度评估的回溯过程，避免了线性算子范数的显式计算。数值模拟结果表明，所提算法相较于现有技术具有显著优势。