New variational principles with applications to optimization theory and algorithms

<p>In this dissertation we investigate some applications of variational analysis in optimization theory and algorithms. In the first part we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, under the name of tangential extremal principles and rated extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. These developments are in the core geometric theory of variational analysis. Our study includes the basic theory and applications to problems of semi-infinite programming and multiobjective optimization. The second part of this dissertation concerns developing numerical methods of the Newton-type to solve systems of nonlinear equations. We propose and justify a new generalized Newton algorithm based on graphical derivatives. Based on advanced tools of variational analysis and generalized differentiation, we establish the well-posedness and convergence results of the algorithm. Besides, we present a new generalized damped Newton algorithm, which is also known as Newton's method with line-search. Some global convergence results are also justified.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2011

Author dc:creator

• Phan, Hung Minh

Contributors dc:contributor

• Boris S. Mordukhovich

• extremal principles
• multiobjective optimization
• Newton's method
• semi-infinite programming
• Mathematics

Repository record dc:identifier

https://digitalcommons.wayne.edu/oa_dissertations/327

OAI identifier oai:identifier

oai:digitalcommons.wayne.edu:oa_dissertations-1326