2020/06/04 by Heinz H. Bauschke, Bauschke, Heinz H., Hui Ouyang +3 · 1 citation
Computer Science · Engineering · Mathematics · #41A50 #41A65 #65B99 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Primary 90C25 #Secondary 46B04 #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2006.02644
openalex publication_date 2020/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from ℝn to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al. proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in ℝn. We generalize their result from ℝn to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces.