AI 中文总结
研究解决单调包含问题，针对现有算法局限，为前向-反射-后向（FRB）算法提出新线搜索策略，保证收敛，扩展算法处理多种算子，经数值实验验证其在加速收敛和计算上比线搜索FBF更具优势。
AI 中文摘要
本文旨在解决一个涉及极大单调算子与连续算子之和的单调包含问题。现有算法在连续算子为余强制或利普希茨连续时可解此问题，但通常需估计全局利普希茨常数，计算成本高且步长限制大。为避免这些局限并处理一般连续算子，采用线搜索子程序。流行的前向-后向-前向（FBF）算法每次迭代需评估连续算子两次，而前向-反射-后向（FRB）算法每次迭代只需评估一次。虽有FRB的线搜索版本，但其仅对局部利普希茨算子保证收敛，对于一般连续算子可能不终止。本文为FRB提出一种新的线搜索策略，即使算子仅连续也能保证收敛，还扩展算法以处理余强制和利普希茨连续算子。最后通过鞍点问题和图像恢复的数值实验表明，带新线搜索的FRB即使在算子为利普希茨连续时也能加速数值收敛，且与线搜索FBF相比有计算优势。
英文摘要
In this article, we aim to solve a monotone inclusion problem involving the sum of a maximally monotone operator and a continuous operator. While several algorithms exist to solve this problem when the continuous operator is cocoercive or Lipschitz continuous, they typically require the estimation of the global Lipschitz constant, which can be computationally expensive and often imposes overly restrictive step-sizes. To avoid these limitations and to handle merely continuous operators, linesearch subroutines are employed. A popular method in this context is the forward-backward-forward (FBF) algorithm (also known as Tseng's splitting), which utilizes a linesearch to guarantee convergence. However, a drawback of FBF is that the continuous operator must be evaluated twice per iteration. On the other hand, the forward-reflected-backward (FRB) algorithm proposed by Malitsky and Tam (2020) requires only a single evaluation of the operator per iteration. Although a linesearch version of FRB exists, its convergence is guaranteed only for locally Lipschitz operators; in fact, we present an example demonstrating that this existing linesearch can fail to terminate when the operator is merely continuous. In this work, we propose a novel linesearch strategy for FRB that is well defined and guarantees convergence even when the operator is merely continuous. We also extend the proposed algorithm to handle additional cocoercive and Lipschitz continuous operators. Finally, we provide numerical experiments on saddle-point problems and image restoration. The numerical results show that FRB with the proposed linesearch can accelerate the numerical convergence even when the operator is Lipschitz continuous. In addition, these results show that the proposed method is competitive with linesearch FBF, offering considerable computational advantages in various scenarios.
