提出一种随机优化新方法，能高效处理带约束的不确定目标函数问题。
A Trust-Region Interior-Point Stochastic Sequential Quadratic Programming Method
- 结合信赖域与内点法，用随机估计构建自适应精度的梯度和目标值
- 在标准假设下证明方法几乎必然收敛到一阶驻点
- 适合求解带确定约束的随机优化问题，如机器学习中的鲁棒建模
本文提出一种信赖域内点随机序贯二次规划（TR-IP-SSQP）方法，用于求解具有随机目标函数及确定性非线性等式与不等式约束的优化问题。在此设定下，目标函数及其梯度的精确值不可得，但可通过随机估计构建。每轮迭代中，该方法构建满足固定概率下自适应精度要求的随机代理算子，以估计目标值与梯度。为处理不等式约束，采用内点法（IPM），其障碍参数按预设衰减序列变化。在标准假设条件下，证明了该方法全局几乎必然收敛至一阶驻点。我们在CUTEst测试集的部分问题及逻辑回归问题上实现该方法，验证其实际性能。
原文摘要 · Abstract (English)
In this paper, we propose a trust-region interior-point stochastic sequential quadratic programming (TR-IP-SSQP) method for solving optimization problems with a stochastic objective and deterministic nonlinear equality and inequality constraints. In this setting, exact evaluations of the objective function and its gradient are unavailable, but their stochastic estimates can be constructed. In particular, at each iteration our method builds stochastic oracles, which estimate the objective value and gradient to satisfy proper adaptive accuracy conditions with a fixed probability. To handle inequality constraints, we adopt an interior-point method (IPM), in which the barrier parameter follows a prescribed decaying sequence. Under standard assumptions, we establish global almost-sure convergence of the proposed method to first-order stationary points. We implement the method on a subset of problems from the CUTEst test set, as well as on logistic regression problems, to demonstrate its practical performance.
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