论文标题
基于局部误差绑定条件的惯性前向后算法的收敛速率
Convergence Rate of Inertial Forward-Backward Algorithms Based on the Local Error Bound Condition
论文作者
论文摘要
“惯性向前折叠算法”（IFB）是一个有力的凸dimoots最小化问题的有力工具，它给出了众所周知的“快速迭代收缩率替代算法”（fista）（fista）（fista），它喜欢$ o \ weft（{\ frac {\ frac {\ frac {k^2} {k^2} {k^2} {k^2} {k^2} {迭代已被证明；通过进行小修改，一个称为“ Fista \ _cd”的加速IFB将功能值的收敛速率提高到$ o \ left（{\ frac {1} {1} {k^2}}} \ right）$，并显示迭代酸盐的弱收敛性。局部误差约束条件对于分析用于解决优化问题的宿主的收敛速率非常有用，并且在实际应用中，大量具有特殊结构的问题通常满足误差绑定条件。自然，使用局部误差绑定条件来得出或提高IFB的收敛速率是一种常见手段。在本文中，基于局部误差约束条件，我们利用了IFB中重要参数$ t_k $的新假设条件，并建立了IFB算法产生的迭代率的提高的融合速率和六$ t_k $满足希尔伯特空间中上述假设条件的迭代元素的强收敛。 It is remarkable that, under the local error bound condition, we establish the strong convergence of the iterates generated by the original FISTA, and prove that the convergence rates of function values for FISTA\_CD is actually related to the value of parameter $a,$ and show that the IFB algorithms with some $t_k$ mentioned above can achieve sublinear convergence rate $o\left( {\ frac {1} {k^p}}} \ right）$对于任何正整数$ p> 1 $。进行了一些数值实验来说明我们的结果。
The "Inertial Forward-Backward algorithm" (IFB) is a powerful tool for convex nonsmooth minimization problems, it gives the well known "fast iterative shrinkage-thresholding algorithm " (FISTA), which enjoys $O\left( {\frac{1}{k^2}} \right)$ global convergence rate of function values, however, no convergence of iterates has been proved; by do a small modification, an accelerated IFB called "FISTA\_CD" improves the convergence rate of function values to $o\left( {\frac{1}{k^2}} \right)$ and shows the weak convergence of iterates. The local error bound condition is extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems, and in practical application, a large number of problems with special structure often satisfy the error bound condition. Naturally, using local error bound condition to derive or improve the convergence rate of IFB is a common means. In this paper, based on the local error bound condition, we exploit an new assumption condition for the important parameter $t_k$ in IFB, and establish the improved convergence rate of function values and strong convergence of the iterates generated by the IFB algorithms with six $t_k$ satisfying the above assumption condition in Hilbert space. It is remarkable that, under the local error bound condition, we establish the strong convergence of the iterates generated by the original FISTA, and prove that the convergence rates of function values for FISTA\_CD is actually related to the value of parameter $a,$ and show that the IFB algorithms with some $t_k$ mentioned above can achieve sublinear convergence rate $o\left( {\frac{1}{k^p}} \right)$ for any positive integer $p>1$. Some numerical experiments are conducted to illustrate our results.
