The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization --- the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) --- belong to this class, and introduce a new algorithm --- the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers.

中文翻译：

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凸优化中的双正交贪婪算法

在凸优化的背景下对贪婪逼近的研究正在成为一个有前途的研究方向，因为贪婪算法正被积极地用于构造关于给定元素集的凸函数的稀疏最小化器。在本文中，我们提出了一种统一的方法来分析某种贪婪型算法，用于最小化 Banach 空间上的凸函数。具体来说，我们为凸优化定义了一类弱双正交贪婪算法，其中包含了广泛的贪婪算法。我们分析了引入的算法类别，并建立了收敛性、收敛速度和数值稳定性的属性，从某种意义上说，允许算法的步骤不精确地执行，但可以控制计算不准确度。我们证明了以下著名的凸优化算法——弱切比雪夫贪婪算法（co）和带有自由松弛的弱贪婪算法（co）——属于这一类，并介绍了一种新算法—— Rescaled Weak Relaxed Greedy Algorithm (co)。所提出的数值实验证明了上述贪心算法在凸最小化设置中的实际性能，与正则化优化相比，这是构造稀疏最小化器的传统方法。