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On the Application of Iterative Methods of Nondifferentiable Optimization to Some Problems of Approximation Theory

机译：关于不可微优化迭代方法在逼近理论中的应用

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摘要

We consider the data fitting problem, that is, the problem of approximating a function of several variables, given by tabulated data, and the corresponding problem for inconsistent (overdetermined) systems of linear algebraic equations. Such problems, connected with measurement of physical quantities, arise, for example, in physics, engineering, and so forth. A traditional approach for solving these two problems is the discrete least squares data fitting method, which is based on discrete l(2)-norm. In this paper, an alternative approach is proposed: with each of these problems, we associate a nondifferentiable (nonsmooth) unconstrained minimization problem with an objective function, based on discrete l(1)-and/or l(infinity)-norm, respectively; that is, these two norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residual using these two norms. Respective subgradients are calculated, and a subgradient method is used for solving these two problems. The emphasis is on implementation of the proposed approach. Some computational results, obtained by an appropriate iterative method, are given at the end of the paper. These results are compared with the results, obtained by the iterative gradient method for the corresponding "differentiable" discrete least squares problems, that is, approximation problems based on discrete l(2)-norm.
机译：我们考虑数据拟合问题，即由列表数据给出的逼近几个变量的函数的问题，以及线性代数方程组不一致（不确定）的相应问题。这些与物理量的测量有关的问题例如在物理，工程等中出现。解决这两个问题的传统方法是离散的最小二乘数据拟合方法，该方法基于离散的l（2）-范数。在本文中，提出了一种替代方法：对于每个这些问题，我们分别基于离散l（1）-和/或l（无限）-范数，将不可微分（非平滑）无约束最小化问题与目标函数相关联。 ;即，将这两个规范用作接近标准。换句话说，正在考虑的问题是通过使用这两个规范使残差最小化而解决的。计算了各自的次梯度，并使用次梯度方法来解决这两个问题。重点是建议方法的实施。通过适当的迭代方法获得的一些计算结果在本文末尾给出。将这些结果与通过迭代梯度法获得的对应“可微”离散最小二乘问题（即基于离散l（2）-范数的近似问题）的结果进行比较。

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