Union averaged operators with applications to proximal algorithms

Abstract:

In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.

Biography: Minh N. Dao received the Ph.D. degree in applied mathematics from the University of Toulouse, France in 2014. He was a Lecturer at Hanoi National University of Education, Vietnam from 2004 to 2010, a Lecturer and Research Assistant at National Institute of Applied Sciences (INSA) in Toulouse, France from 2013 to 2014, and a Postdoctoral Fellow at the University of British Columbia, Canada from 2014 to 2016. He is currently a Research Associate in the Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) at The University of Newcastle, Australia. His research interests include nonlinear optimization, nonsmooth analysis, iterative methods, monotone operator theory, control theory, and operations research. In 2017, he received the Annual Best Paper Award from the Journal of Global Optimization.