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.

在凸优化的背景下研究贪心逼近正成为一个有前途的研究方向，因为贪心算法正被积极地用于针对给定的元素集构造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，以最小化 Banach 空间上的凸函数。具体来说，我们定义了用于凸优化的弱双正交贪心算法类，其中包含各种贪心算法。我们分析了引入的算法类别，并确定了收敛、收敛速度和数值稳定性的属性，从某种意义上说，算法的步骤允许不精确地执行，而是在受控的计算不准确性的情况下执行。我们展示了以下著名的凸优化算法——Weak Chebyshev Greedy Algorithm (co) 和 Weak Greedy Algorithm with Free Relaxation (co)——属于这一类，并介绍了一种新算法——Rescaled Weak Relaxed Greedy Algorithm （合）。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，正则化是构造稀疏最小化器的传统方法。