On a manifold or a closed subset of a Euclidean vector space, a retraction enables to move in the direction of a tangent vector while staying on the set. Retractions are a versatile tool to perform computational tasks such as optimization, interpolation, and numerical integration. This paper studies two definitions of retraction on a closed subset of a Euclidean vector space, one being weaker than the other. Specifically, it shows that, in the context of constrained optimization, the weaker definition should be preferred as it inherits the main property of the other while being less restrictive.

翻译：在欧几里得向量空间的流形或闭子集上，回缩使得沿切向量方向移动时仍能保持在该集合内。回缩是执行优化、插值和数值积分等计算任务的通用工具。本文研究了欧几里得向量空间闭子集上回缩的两种定义，其中一种比另一种更弱。具体而言，研究表明，在约束优化的背景下，应优先采用较弱的定义，因其在限制条件更少的同时继承了较强定义的主要性质。