针对噪声环境下的等式约束优化，提出改进的信赖域算法。
A Trust-Region Algorithm for Noisy Equality Constrained Optimization
- 引入噪声估计来决定步长接受与信赖域更新
- 证明迭代点收敛到与噪声水平相关的驻点区域
- 适合处理带噪声的大型等式约束优化问题
本文提出一种改进的Byrd-Omojokun（BO）信赖域算法，以应对函数和梯度评估存在噪声时的优化挑战。原BO方法适用于等式约束问题，是某些大规模约束优化内点法的核心。其关键优势在于能处理约束雅可比矩阵秩亏的情况。本文提出的算法引入新的步长接受准则和信赖域更新机制，利用对问题噪声水平的估计。分析表明，在特定条件下，迭代点会收敛至由噪声水平决定的驻点邻域。该分析比线搜索方法更复杂，因信赖域包含前次迭代的（噪声）信息。数值实验验证了算法的实际性能。
原文摘要 · Abstract (English)
This paper introduces a modified Byrd-Omojokun (BO) trust region algorithm to address the challenges posed by noisy function and gradient evaluations. The original BO method was designed to solve equality constrained problems and it forms the backbone of some interior point methods for general large-scale constrained optimization. A key strength of the BO method is its robustness in handling problems with rank-deficient constraint Jacobians. The algorithm proposed in this paper introduces a new criterion for accepting a step and for updating the trust region that makes use of an estimate in the noise in the problem. The analysis presented here gives conditions under which the iterates converge to regions of stationary points of the problem, determined by the level of noise. This analysis is more complex than for line search methods because the trust region carries (noisy) information from previous iterates. Numerical tests illustrate the practical performance of the algorithm.
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