2021/05/05 by Hui Ouyang, Ouyang, Hui · 1 citation
Computer Science · Mathematics · #47H05 #49M37 #52A41 #65K05 #FOS: Mathematics #Fixed Point Theorems Analysis #Mathematical Inequalities and Applications #Optimization and Control (math.OC) #Optimization and Variational Analysis #Primary 90C48 #Secondary 90C25
paper · pdf · doi:10.48550/arxiv.2105.02308
openalex publication_date 2021/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, we systematically studied the basic theory of Bregman circumcenters in another paper. In this work, we aim to apply Bregman circumcenters to optimization algorithms. Here, we propose the forward Bregman monotonicity which is a generalization of the powerful Fejér monotonicity and show a weak convergence result of the forward Bregman monotone sequence. We also naturally introduce the Bregman circumcenter mappings associated with a finite set of operators. Then we provide sufficient conditions for the sequence of iterations of the forward Bregman circumcenter mapping to be forward Bregman monotone. Furthermore, we prove that the sequence of iterations of the forward Bregman circumcenter mapping weakly converges to a point in the intersection of the fixed point sets of relevant operators, which reduces to the known weak convergence result of the circumcentered method under the Euclidean distance. In addition, particular examples are provided to illustrate the Bregman isometry and Browder's demiclosedness principle, and our convergence result.