Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost.

中文翻译：

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关于非线性规划的增广拉格朗日策略的收敛性

增强拉格朗日 (AL) 算法是解决约束优化问题的非常流行和成功的方法。最近，通过使用顺序最优条件的概念，这些方法的全局收敛性分析得到了显着改进。无论是否满足任何约束条件，这些条件对于最优性都是必要的，并提供理论工具来证明几种数值优化方法的停止标准是合理的。在这里，我们引入了一个比文献中先前陈述的更强的新的顺序最优性条件。我们展示了一个完善的受保护的 Powell-Hestenes-Rockafellar (PHR) AL 算法在 Lojasiewicz 型假设下生成满足新条件的点，改进和统一了所有先前的收敛结果。此外，我们引入了一种新的原始对偶 AL 方法，能够在没有 Lojasiewicz 假设的情况下实现这些点。然后，我们提出了一种混合方法，其中新策略在受保护的 PHR 方法趋于失败时起到帮助作用。我们通过初步数值测试表明，已通过受保护的 PHR 方法成功解决的所有问题保持不变，而 PHR 方法失败的其他问题现在以可接受的额外计算成本解决。