SIAM Review, Volume 65, Issue 2, Page 537-537, May 2023.
The Education section in this issue presents two contributions. In `"Nesterov's Method for Convex Optimization," Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, "A Comprehensive Proof of Bertrand's Theorem," is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.

中文翻译：

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教育

SIAM Review，第 65 卷，第 2 期，第 537-537 页，2023 年 5 月。
本期的教育部分提出了两项​​贡献。在“Nesterov 的凸优化方法”中，Noel J. Walkington 提出了第一门课程的教学指南，用于优化这种计算凸函数最小值的著名算法。该算法于 1983 年由 Yuri Nesterov 首次提出，虽然在存在大数据的计算优化中被公认为比最速下降法更有效的工具，但在大多数现代优化教科书中仍然没有出现。本文的作者对 Nesterov 的一阶算法进行了基本分析，该算法与最速下降法相似，但具有 Nesterov 提出的附加要求。讨论了两种情况。第一个涉及无约束最小化问题，而第二个包括使用成本的无限惩罚表示的封闭凸约束。更一般地，成本函数变成光滑凸函数和下半连续凸函数的总和。本文包括几个学生级别的练习。信号恢复问题的示例和 Uzawa 算法的讨论很好地说明了结果，该优化问题具有由涉及凸函数的不等式定义的约束。第二篇论文“Bertrand 定理的综合证明”由 Patrick De Leenheer、John Musgrove 和 Tyler Schimleck 发表。它涉及经典二体问题的解的行为，并指出，在所有可能的引力定律中，只有两个表现出所有有界轨道都是封闭的性质：牛顿万有引力和胡克万有引力。从历史上看，即使牛顿知道存在具有上述性质的特定引力定律，但仅在两个世纪后的 1873 年，伯特兰才意识到只有这些。伯特兰定理由于其重要的后果，已被纳入理论力学的本科课程，但现代教科书中似乎没有它的证明，本科数学或物理学生可以获得。尽管 Bertrand 的原始论文没有包含精确的证明，但 V. Arnold 提出了基于六个子问题的草图。在其他贡献中，本文在对运动模型中力的大小施加特定假设的情况下提供了第六个子问题的完整证明。在这个假设下，然后给出 Bertrand 定理的完整证明，其中还包含其他作者的早期贡献。尽管如此，这里的全面并不意味着简单，这篇论文可以用来构思几个数学或物理学高年级本科生的研究项目。