AI 中文总结
本文提出两种无参数及自适应Halpern不动点算法，结合Tikhonov正则化得到求解余强制方程的无参数方法，还推导了无参数Nesterov加速变体，均具线性收敛性，实验显示其性能优于或相当现有自适应不动点方法。
AI 中文摘要
本文针对压缩映射开发了可证明无参数且自适应的不动点算法，重点在于自动利用隐藏的压缩性，无需预先知晓压缩因子。第一种方法是完全无参数的Halpern不动点迭代变体，它不需要线搜索、二分法或压缩因子的先验估计，同时保留了与经典不动点方案基本相同的每次迭代计算成本。我们为不动点残差和到唯一不动点的距离建立了明确的线性收敛速率。第二种算法是自适应Halpern方法，仅需要压缩因子的上界，在非扩张情况下可简化为现有自适应Halpern方案，该方法也具有明确的线性收敛保证。我们进一步将这些思路扩展到两个方向：一是将所提出的不动点方案与Tikhonov正则化相结合，得到求解余强制方程的无参数方法，并建立了计算ε-解的迭代复杂度为𝒪(ε⁻¹ln(ε⁻¹))；二是利用Halpern迭代与Nesterov加速不动点方案之间的关系，推导了无参数的Nesterov加速变体，在压缩设置中继承了线性收敛性。对多个示例的数值实验表明，所提算法与现有自适应不动点方法相比具有竞争力，且通常表现更优，特别是在存在压缩行为时能成功利用该特性，同时对非扩张问题仍保持有效性。
英文摘要
In this paper, we develop provable parameter-free and adaptive fixed-point algorithms for contractive mappings, with an emphasis on automatically exploiting hidden contractivity without requiring prior knowledge of the contraction factor. Our first method is a completely parameter-free variant of the Halpern fixed-point iteration. It requires no line search, bisection, or prior estimate of the contraction factor, while retaining essentially the same per-iteration computational cost as classical fixed-point schemes. We establish explicit linear convergence rates for both the fixed-point residual and the distance to the unique fixed point. The second algorithm is an adaptive Halpern method that requires only an upper bound on the contraction factor and reduces to an existing adaptive Halpern scheme in the nonexpansive case. This method also enjoys explicit linear convergence guarantees. We further extend these ideas in two directions. First, by combining the proposed fixed-point schemes with Tikhonov regularization, we obtain a parameter-free method for solving co-coercive equations and establish an iteration complexity of $\mathcal{O}({ε^{-1}\ln(ε^{-1})})$ for computing an $ε$-solution. Second, using the relation between Halpern iterations and Nesterov's accelerated fixed-point schemes, we derive parameter-free Nesterov's accelerated variants that inherit linear convergence in the contractive setting. Numerical experiments on several examples demonstrate that the proposed algorithms are competitive with, and often outperform, existing adaptive fixed-point methods. In particular, the methods successfully exploit contractive behavior when it is present while remaining effective on nonexpansive problems.
Comments28 pages, 6 pigures, and 4 tables
