1. Bregman Circumcenters: Basic Theory
- Author
-
Hui Ouyang and Xianfu Wang
- Subjects
Control and Optimization ,Explicit formulae ,Iterative method ,Duality (mathematics) ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Computer Science::Computational Geometry ,Management Science and Operations Research ,Bregman divergence ,01 natural sciences ,Legendre function ,Statistics::Machine Learning ,Development (topology) ,FOS: Mathematics ,Mathematics::Metric Geometry ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Primary 90C48, 47H04, 47H05, Secondary 90C25, 52A41 ,Finite set ,Mathematics ,021103 operations research ,Applied Mathematics ,Optimization and Control (math.OC) ,Theory of computation - Abstract
Circumcenters play an important role in the design and analysis of accelerating various iterative methods in optimization. In this work, we propose Bregman (pseudo-)circumcenters associated with finite sets. We show the existence and give explicit formulae for the unique backward and forward Bregman pseudo-circumcenters of finite sets. Moreover, we use duality to establish connections between backward and forward Bregman (pseudo-)circumcenters. Various examples are presented to illustrate the backward and forward Bregman (pseudo-)circumcenters of finite sets. Our general framework for circumcenters paves the way for the development of accelerating iterative methods by Bregman circumcenters., Comment: 24 pages and 4 figures
- Published
- 2021