In this paper, we consider nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method and establish its high-probability iteration complexity bounds for identifying first- and second-order $\epsilon$-stationary points. In our algorithm, we assume that exact objective values, gradients, and Hessians are not directly accessible but can be estimated via zeroth-, first-, and second-order probabilistic oracles. Compared to existing complexity studies of SSQP methods that rely on a zeroth-order oracle with sub-exponential tail noise (i.e., light-tailed) and focus mostly on first-order stationarity, our analysis accommodates irreducible and heavy-tailed noise in the zeroth-order oracle and significantly extends the analysis to second-order stationarity. We show that under heavy-tailed noise conditions, our SSQP method achieves the same high-probability first-order iteration complexity bounds as in the light-tailed noise setting, while further exhibiting promising second-order iteration complexity bounds. Specifically, the method identifies a first-order $\epsilon$-stationary point in $\mathcal{O}(\epsilon^{-2})$ iterations and a second-order $\epsilon$-stationary point in $\mathcal{O}(\epsilon^{-3})$ iterations with high probability, provided that $\epsilon$ is lower bounded by a constant determined by the irreducible noise level in estimation. We validate our theoretical findings and evaluate the practical performance of our method on CUTEst benchmark test set.

翻译：本文研究了具有随机目标函数和确定性等式约束的非线性优化问题。我们提出了一种信任域随机序列二次规划方法，并建立了该方法识别一阶和二阶$\epsilon$-平稳点的高概率迭代复杂度界。在我们的算法中，假设无法直接获取精确的目标函数值、梯度和Hessian矩阵，但可以通过零阶、一阶和二阶概率预言机进行估计。与现有依赖具有次指数尾噪声（即轻尾）的零阶预言机且主要关注一阶平稳性的SSQP方法复杂度研究相比，我们的分析兼容了零阶预言机中不可约且重尾的噪声，并将分析显著扩展至二阶平稳性。我们证明，在重尾噪声条件下，我们的SSQP方法实现了与轻尾噪声设置相同的高概率一阶迭代复杂度界，同时进一步展现出有前景的二阶迭代复杂度界。具体而言，只要$\epsilon$不低于由估计中不可约噪声水平决定的常数，该方法以高概率在$\mathcal{O}(\epsilon^{-2})$次迭代内识别一阶$\epsilon$-平稳点，在$\mathcal{O}(\epsilon^{-3})$次迭代内识别二阶$\epsilon$-平稳点。我们在CUTEst基准测试集上验证了理论发现并评估了方法的实际性能。