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 $ε$-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 biased (also referred to as irreducible in the literature) 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 $ε$-stationary point in $\mathcal{O}(ε^{-2})$ iterations and a second-order $ε$-stationary point in $\mathcal{O}(ε^{-3})$ iterations with high probability, provided that $ε$ is lower bounded by a constant determined by the bias magnitude (i.e., the irreducible noise) in the estimation. We validate our theoretical findings and evaluate practical performance of our method on CUTEst benchmark test set.

翻译：本文考虑具有随机目标函数和确定性等式约束的非线性优化问题。我们提出一种信赖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ε-驻点的高概率迭代复杂度界。在算法中，我们假设无法直接获取精确的目标值、梯度和海森矩阵，但可通过零阶、一阶和二阶概率预言机进行估计。与现有依赖于具有次指数尾部噪声（即轻尾）且主要关注一阶驻点性的SSQP方法复杂度研究相比，我们的分析可容纳零阶预言机中的有偏（文献中也称为不可约噪声）和重尾噪声，并将分析显著扩展至二阶驻点性。我们证明在重尾噪声条件下，所提出的SSQP方法实现了与轻尾噪声设定相同的高概率一阶迭代复杂度界，同时进一步展现出具有前景的二阶迭代复杂度界。具体而言，该方法能以高概率在O(ε⁻²)次迭代内识别一阶ε-驻点，在O(ε⁻³)次迭代内识别二阶ε-驻点，条件为ε被估计中偏差幅度（即不可约噪声）决定的常数所下界约束。我们在CUTEst基准测试集上验证了理论结果并评估了方法的实际性能。