We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.

我们引入了一种重构技术，将多集可行性问题转换为等效的二集问题。该技术包括在应用 Pierra 的经典乘积空间重构之前，通过用它们的交集替换其约束集对来重构原始可行性问题。合并两个约束集的步骤减少了乘积空间的维数。我们将这种技术称为约束减少重构并使用它来获得众所周知的投影算法的约束减少变体，例如 Douglas-Rachford 算法和交替投影方法等。我们证明了在非凸设置中存在凸性和局部收敛的情况下约束减少算法的全局收敛。为了分析约束减少的 Douglas-Rachford 方法的收敛性，我们概括了一个经典结果，该结果保证了两个投影在子空间上的组合是在它们的交点上的投影。最后，我们应用 Douglas-Rachford 和交替投影的约束减少版本来解决小波可行性问题，然后将它们的性能与其通常的产品变体进行比较。