In this work, we consider solving optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Sequential Quadratic Programming method to find both first- and second-order stationary points. Our method utilizes a random model to represent the objective function, which is constructed from stochastic observations of the objective and is designed to satisfy proper adaptive accuracy conditions with a high but fixed probability. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a quadratic approximation of the objective subject to a (relaxed) linear approximation of the problem constraints and a trust-region constraint. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature of the reduced Hessian matrix, as well as a second-order correction step to address the potential Maratos effect, which arises due to the nonlinearity of the problem constraints. Such an effect may impede the method from moving away from saddle points. Both gradient and eigen step computations leverage a novel parameter-free decomposition of the step and the trust-region radius, accounting for the proportions among the feasibility residual, optimality residual, and negative curvature. We establish global almost sure first- and second-order convergence guarantees for our method, and present computational results on CUTEst problems, regression problems, and saddle-point problems to demonstrate its superiority over existing line-search-based stochastic methods.

翻译：本文研究具有随机目标函数和确定性等式约束的优化问题求解。我们提出一种信赖域序列二次规划方法，用于寻找一阶和二阶稳定点。该方法采用随机模型表示目标函数，该模型通过目标函数的随机观测构建，并以高固定概率满足适当的自适应精度条件。为收敛至一阶稳定点，本方法在每次迭代中通过最小化目标函数的二次近似（在问题约束的（松弛）线性近似及信赖域约束条件下）计算梯度步。为收敛至二阶稳定点，本方法额外计算特征步以探索约化Hessian矩阵的负曲率，同时引入二阶校正步以应对可能出现的Maratos效应——该效应源于问题约束的非线性特性，可能阻碍算法逃离鞍点。梯度步与特征步的计算均采用创新的无参数分解策略，将步长与信赖域半径按可行性残差、最优性残差及负曲率之间的比例关系进行分解。我们建立了该方法全局几乎必然的一阶与二阶收敛性保证，并在CUTEst测试集、回归问题及鞍点问题上展示了计算效果，验证其优于现有基于线搜索的随机方法。