等式约束下二阶点的计算：重访Fletcher增广拉格朗日方法
Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian
AI总结：
本文在光滑等式约束优化中，基于黎曼优化几何形式定义近似二阶临界点，并利用Fletcher增广拉格朗日提出算法，以O(ε^{-3})迭代达到ε-近似二阶临界点，改进了现有理论界。
摘要：
我们研究在光滑等式约束下最小化光滑函数的问题。在约束的正则性假设下，我们基于黎曼优化的几何形式，提出了一种近似一阶和二阶临界点的概念。利用称为Fletcher增广拉格朗日的光滑精确罚函数，我们提出了一种最小化罚成本函数的算法，该算法在至多$\mathcal{O}(\varepsilon^{-3})$次迭代内达到原优化问题的$\varepsilon$-近似二阶临界点。这改进了当前最好的理论界。在此过程中，我们展示了Fletcher增广拉格朗日的新性质，这些性质可能具有独立的研究价值。
英文摘要：
We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches $\varepsilon$-approximate second-order critical points of the original optimization problem in at most $\mathcal{O}(\varepsilon^{-3})$ iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest.
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