We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where at each step a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to utilize indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm must address an infeasibility issue -- the linearized equality constraints and trust-region constraints may lead to infeasible SQP subproblems. In this regard, we propose an adaptive relaxation technique to compute the trial step, consisting of a normal step and a tangential step. To control the lengths of these two steps while ensuring a scale-invariant property, we adaptively decompose the trust-region radius into two segments, based on the proportions of the rescaled feasibility and optimality residuals to the rescaled full KKT residual. The normal step has a closed form, while the tangential step is obtained by solving a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish a global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.

翻译：我们提出了一种信赖域随机序列二次规划算法（TR-StoSQP），用于求解具有随机目标函数和确定性等式约束的非线性优化问题。我们考虑完全随机设置，其中每一步仅生成单个样本以估计目标梯度。该算法自适应选择信赖域半径，与现有的线搜索StoSQP方案相比，允许在SQP子问题中使用不定黑塞矩阵（即无需修正的黑塞矩阵）。作为针对约束优化的信赖域方法，我们的算法必须解决一个不可行性问题——线性化等式约束和信赖域约束可能导致SQP子问题不可行。为此，我们提出了一种自适应松弛技术来计算试验步，该步由法向步和切向步组成。为了在确保尺度不变特性的同时控制这两个步长，我们根据重新缩放后的可行性和最优性残差与重新缩放后的完整KKT残差的比例，自适应地将信赖域半径分解为两个部分。法向步具有封闭形式的解，而切向步则通过求解一个信赖域子问题获得，其中确保柯西约化的解对于我们的研究已足够。我们建立了TR-StoSQP的全局几乎必然收敛性保证，并通过CUTEst测试集中的部分问题以及使用LIBSVM数据集集合中的数据进行约束逻辑回归问题，展示了其经验性能。