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Variational inclusions with general over-relaxed proximal point and variational-like inequalities with densely pseudomonotonicity

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Date Issued:
2019
Abstract/Description:
This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ? ? ? pseudomonotone operators and generalized relaxed ? ? ? proper quasimonotone operators as well as relaxed ? ? ? quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work.
Title: Variational inclusions with general over-relaxed proximal point and variational-like inequalities with densely pseudomonotonicity.
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Name(s): Nguyen, George, Author
Mohapatra, Ram, Committee Chair
Han, Deguang, Committee Member
Shuai, Zhisheng, Committee Member
Xu, Mengyu, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2019
Publisher: University of Central Florida
Language(s): English
Abstract/Description: This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ? ? ? pseudomonotone operators and generalized relaxed ? ? ? proper quasimonotone operators as well as relaxed ? ? ? quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work.
Identifier: CFE0007693 (IID), ucf:52410 (fedora)
Note(s): 2019-08-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): variational inclusion -- variational inequality
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007693
Restrictions on Access: public 2019-08-15
Host Institution: UCF

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