论文标题
在平等约束下计算二阶的计算点：重新访问Fletcher的增强Lagrangian
Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian
论文作者
论文摘要
我们解决了在平滑平等限制下最小化光滑函数的问题。在这些约束的规律性假设下，我们提出了一个近似一阶和二阶临界点的概念，该概念依赖于黎曼优化的几何形式主义。使用称为Fletcher的增强拉格朗日的平滑惩罚功能，我们提出了一种算法，以最大程度地减少罚款成本功能，该惩罚成本功能达到$ \ varepsilon $ - 在最多$ \ Mathcal {O}（O}（O}（O}（\ varepsilon^{\ varepsilon^{ - 3}} { - 3}}））$ ISERIAN $ MATHCAL PROUTION PLICTION OFICATION COMPATION的二阶关键点。这改善了当前最佳理论界限。一路上，我们展示了弗莱彻（Fletcher）增强的拉格朗日（Lagrangian）的新属性，这可能具有独立的兴趣。
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.
