论文标题
螺旋算法及其双重的可计算核心方法，并具有Lyapunov函数理论的动机
Computable Centering Methods for Spiraling Algorithms and their Duals, with Motivations from the theory of Lyapunov Functions
论文作者
论文摘要
对于许多问题（其中一些问题）在论文中进行了审查，如Douglas-Rachford（DR），ADMM和FISTA这样的流行算法会产生近似序列，这些序列显示出朝向溶液螺旋的迹象。我们提出了一个元算法，该元算法利用了这种动态来潜在提高性能。这种元偏金属的策略是迭代构建和最小化替代这些动态的Lyapunov函数的替代物。 作为第一个激励应用，我们表明，对于典型的可行性问题，围绕的反射方法（CRM），亚甘型预测和牛顿 - 拉夫森都可以描述为基于梯度的方法，用于最小化为DR操作员构建的Lyapunov功能，而前者则返回lyapunov for lyapunov forlyapunov forly for lyapunov funcorial的最小化。 作为第二个激励应用程序，我们引入了一种新方法，该方法共享这些属性，但具有更多优势：1）不依赖子问题（例如反射），因此可以适用于任何迭代型具有螺旋属性的操作员； 2）事实证明，上述的Lyapunov属性很少，结构性假设很少，因此通常适合原始/双重实施； 3）每当原始运算符时，将尺寸缩小的尺寸缩小空间。这使得寻求螺旋式迭代中心的方法的第一个原始/双重实现可能。我们描述了此方法，并提供了一个计算的示例（基础追求）。
For many problems, some of which are reviewed in the paper, popular algorithms like Douglas--Rachford (DR), ADMM, and FISTA produce approximating sequences that show signs of spiraling toward the solution. We present a meta-algorithm that exploits such dynamics to potentially enhance performance. The strategy of this meta-algorithm is to iteratively build and minimize surrogates for the Lyapunov function that captures those dynamics. As a first motivating application, we show that for prototypical feasibility problems the circumcentered-reflection method (CRM), subgradient projections, and Newton--Raphson are all describable as gradient-based methods for minimizing Lyapunov functions constructed for DR operators, with the former returning the minimizers of spherical surrogates for the Lyapunov function. As a second motivating application, we introduce a new method that shares these properties but with the added advantages that it: 1) does not rely on subproblems (e.g. reflections) and so may be applied for any operator whose iterates have the spiraling property; 2) provably has the aforementioned Lyapunov properties with few structural assumptions and so is generically suitable for primal/dual implementation; and 3) maps spaces of reduced dimension into themselves whenever the original operator does. This makes possible the first primal/dual implementation of a method that seeks the center of spiraling iterates. We describe this method, and provide a computed example (basis pursuit).
