Abstract Subspace properties are presented for the trust-region subproblem that appears in the Augmented Lagrangian-Trust-Region method recently proposed by Wang and Yuan (2015). Specifically, when the approximate Lagrangian Hessians are updated by suitable quasi-Newton formulas, we show that any solution of the corresponding kth subproblem belongs to the subspace spanned by all gradient vectors of the objective and of the constraints computed up to iteration k. From this result, a subspace version of the referred method is proposed for large-scale equality constrained optimization problems. The subspace method is suitable to problems in which the number of constraints is much lower than the number of variables.

中文翻译：

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用于等式约束优化的 Wang-Yuan Augmented Lagrangian-Trust Region 方法的子空间版本

摘要 针对最近由 Wang 和 Yuan (2015) 提出的增强拉格朗日-信任区域方法中出现的信任区域子问题，提出了子空间属性。具体来说，当近似的拉格朗日 Hessian 由合适的拟牛顿公式更新时，我们表明对应第 k 个子问题的任何解都属于由目标的所有梯度向量和直到迭代 k 计算的约束所跨越的子空间。根据该结果，针对大规模等式约束优化问题提出了所提及方法的子空间版本。子空间方法适用于约束数量远低于变量数量的问题。