We analyze the complexity of single-loop quadratic penalty and augmented Lagrangian algorithms for solving nonconvex optimization problems with functional equality constraints. We consider three cases, in all of which the objective is stochastic and smooth, that is, an expectation over an unknown distribution that is accessed by sampling. The nature of the equality constraints differs among the three cases: deterministic and linear in the first case, deterministic, smooth and nonlinear in the second case, and stochastic, smooth and nonlinear in the third case. Variance reduction techniques are used to improve the complexity. To find a point that satisfies $\varepsilon$-approximate first-order conditions, we require $\widetilde{O}(\varepsilon^{-3})$ complexity in the first case, $\widetilde{O}(\varepsilon^{-4})$ in the second case, and $\widetilde{O}(\varepsilon^{-5})$ in the third case. For the first and third cases, they are the first algorithms of "single loop" type (that also use $O(1)$ samples at each iteration) that still achieve the best-known complexity guarantees.

分析具有函数等式约束的非凸优化问题的单环二次罚函数和增广Lagrange算法的复杂性，通过三个不同情况的考虑得出具体复杂度，使用方差减少技术改进复杂度，这些算法为第一种和第三种情况的首个“单环”类型算法并且仍然具有最佳已知复杂度保证。

随机目标和约束下非线性规划的单循环算法复杂性