Iterative algorithms aimed at solving some problems are discussed. For certain problems, such as finding a common point in the intersection of a finite number of convex sets, there often exist iterative algorithms that impose very little demand on computer resources. For other problems, such as finding that point in the intersection at which the value of a given function is optimal, algorithms tend to need more computer memory and longer execution time. A methodology is presented whose aim is to produce automatically for an iterative algorithm of the first kind a "superiorized version" of it that retains its computational efficiency but nevertheless goes a long way towards solving an optimization problem. This is possible to do if the original algorithm is "perturbation resilient," which is shown to be the case for various projection algorithms for solving the consistent convex feasibility problem. The superiorized versions of such algorithms use perturbations that steer the process in the direction of a superior feasible point, which is not necessarily optimal, with respect to the given function. After presenting these intuitive ideas in a precise mathematical form, they are illustrated in image reconstruction from projections for two different projection algorithms superiorized for the function whose value is the total variation of the image.

中文翻译：

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迭代算法的扰动弹性和优越性

讨论了旨在解决一些问题的迭代算法。对于某些问题，例如在有限数量的凸集的交集上寻找公共点，通常存在对计算机资源要求很少的迭代算法。对于其他问题，例如找到给定函数值最佳的交集点，算法往往需要更多的计算机内存和更长的执行时间。提出了一种方法，其目的是为第一类迭代算法自动生成它的“高级版本”，该算法保留其计算效率，但仍然在解决优化问题方面有很长的路要走。如果原始算法具有“扰动弹性”，则可以这样做，对于解决一致凸可行性问题的各种投影算法来说，情况就是如此。此类算法的高级版本使用扰动来引导过程朝着优越的可行点的方向发展，这对于给定的函数来说不一定是最佳的。在以精确的数学形式呈现这些直观的想法之后，它们在根据两种不同的投影算法的投影重建图像中得到说明，这些算法的优势在于其值是图像的总变化的函数。