On Bauschke-Bendit-Moursi modulus of averagedness and generalized monotone operators
department or school: Science, Irving K. Barber Faculty of (Okanagan)
department or school: Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
data provider: DSpace
degree campus: UBCO
creator: Song, Shuang
issued: 2025-01-03T23:05:16Z
issued: 2024
related degree: Master of Science - MSc
degree grantor: University of British Columbia
description: Averaged operators are important in convex analysis and optimization algorithms. In the first part of this thesis, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators by using the Bauschke-Bendit-Moursi modulus of averagedness. We show that if an averaged operator has modulus of averagedness less than 1/2, then it is a bi-Lipschitz homeomorphism. Amazingly the proximity operator of a convex function has its modulus of averagedness less than 1/2 if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximity operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results. In the second part of this thesis, we focus on a natural generalization of averaged operators and monotonicity. We extend the definition of modulus of averagedness and show how it behaves for linear operators. As an application, explicit formulae are obtained with respect to angle between subspaces. We will also present various profound nonlinear results about this generalization of averaged operator and monotonicity. All the results can apply to averaged operators. Examples are provided to illustrate our results.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/90041
note: On Bauschke-Bendit-Moursi modulusof averagedness and generalizedmonotone operatorsbyShuang SongB.Sc., Sichuan University, 2023A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2024© Shuang Song, 2024The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:On Bauschke-Bendit-Moursi modulus of averagedness and gen-eralized monotone operatorssubmitted by Shuang Song in partial fulfilment of the requirements of thedegree of Master of ScienceDr. Shawn X. Wang, Irving K. Barber Faculty of ScienceSupervisorDr. Heinz Bauschke, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Amy Wiebe, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Stephen Karrer O’Leary, School of EngineeringUniversity ExamineriiAbstractAveraged operators are important in convex analysis and optimizationalgorithms. In the first part of this thesis, we propose classifications of av-eraged operators, firmly nonexpansive operators, and proximal operators byusing the Bauschke-Bendit-Moursi modulus of averagedness. We show thatif an averaged operator has modulus of averagedness less than 1/2, then it isa bi-Lipschitz homeomorphism. Amazingly the proximity operator of a con-vex function has its modulus of averagedness less than 1/2 if and only if thefunction is Lipschitz smooth. Some results on the averagedness of operatorcompositions are obtained. Explicit formulae for calculating the modulusof averagedness of resolvents and proximity operators in terms of variousvalues associated with the maximally monotone operator or subdifferentialare also given. Examples are provided to illustrate our results.In the second part of this thesis, we focus on a natural generalizationof averaged operators and monotonicity. We extend the definition of mod-ul…
has type: Thesis/Dissertation
date issued: 2025-02
is shown at: 10.14288/1.0447668
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Wang, Xianfu
type: Text
identifier URI: http://hdl.handle.net/2429/90041