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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Phan, Hung Minh"],"institution":null,"degree_name":"Ph.D.","degree_level":"Open Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boris S. Mordukhovich"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-01T08:00:00Z","date_published":"2011-01-01T08:00:00Z","updated_at":"2026-07-24T05:58:57Z","subjects":["extremal principles","multiobjective optimization","Newton's method","semi-infinite programming","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wayne.edu/oa_dissertations/327","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boris S. Mordukhovich"]},{"key":"dc:creator","label":"Author","values":["Phan, Hung Minh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2011-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["extremal principles","multiobjective optimization","Newton's method","semi-infinite programming","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wayne.edu/oa_dissertations/327"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["New variational principles with applications to optimization theory and algorithms"]}]}],"canonical_facts":{"dc:contributor":["Boris S. Mordukhovich"],"dc:creator":["Phan, Hung Minh"],"dc:date.available":["2011-01-01T08:00:00Z"],"dc:description.abstract":["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.
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