{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72533"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72533","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Theory and Algorithms for Nonlinear Optimization and Variational Inequalities","abstract":"We consider three separate topics in nonlinear optimization, one theoretical topic and two algorithmic topics. Each of these topics deals with a broad class of nonlinear optimization problems. We first introduce and analyze a generalized parametric variational inequality problem $PVI(E, T, C, \\psi$, Z) in locally convex Hausdorff topological vector spaces. Based on Nikaido's coincidence theorem, we establish several general existence theorems, even without requiring convexity nor contractibility on T and C, but merely a certain acyclic property. The case where C is not compact is considered. We also analyze asymptotic convergence of the partial proximal point algorithm for solving the generalized nonlinear equation $0\\in T(x),$ where $T : H\\to H$ is a maximal monotone multifunction, and $H := H\\sb1 \\times H\\sb2$ is a product of two real Hilbert spaces. Under the mild feasibility assumption $O\\in int(coR(T)),$ we show that the partial proximal point algorithm has the same convergence properties as does Rockafellar's proximal point algorithm. Moreover, the partial convergence rates are shown to depend upon how rapidly $T\\sp{-1}$ grows away from the solution set $T\\sb{-1}(0).$ The partial rate of convergence is linear, superlinear, or quadratic, depending upon certain parameters. Finally, we describe a primal-dual path-following interior point algorithm for solving the pair of primal-dual problems (P) and (D):","abstract_html":"We consider three separate topics in nonlinear optimization, one theoretical topic and two algorithmic topics. Each of these topics deals with a broad class of nonlinear optimization problems. We first introduce and analyze a generalized parametric variational inequality problem $PVI(E, T, C, \\psi$, Z) in locally convex Hausdorff topological vector spaces. Based on Nikaido's coincidence theorem, we establish several general existence theorems, even without requiring convexity nor contractibility on T and C, but merely a certain acyclic property. The case where C is not compact is considered. We also analyze asymptotic convergence of the partial proximal point algorithm for solving the generalized nonlinear equation $0\\in T(x),$ where $T : H\\to H$ is a maximal monotone multifunction, and $H := H\\sb1 \\times H\\sb2$ is a product of two real Hilbert spaces. Under the mild feasibility assumption $O\\in int(coR(T)),$ we show that the partial proximal point algorithm has the same convergence properties as does Rockafellar's proximal point algorithm. Moreover, the partial convergence rates are shown to depend upon how rapidly $T\\sp{-1}$ grows away from the solution set $T\\sb{-1}(0).$ The partial rate of convergence is linear, superlinear, or quadratic, depending upon certain parameters. Finally, we describe a primal-dual path-following interior point algorithm for solving the pair of primal-dual problems (P) and (D):","abstract_has_math":true,"creators":["Chu, Liang-Ju"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["McLinden, L.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:44Z","date_published":"2014-12-17T23:17:44Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9305491"],"render_values":[{"text":"(UMI)AAI9305491","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72533","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McLinden, L.,"]},{"key":"dc:creator","label":"Author","values":["Chu, Liang-Ju"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:44Z","10000-01-01","1992"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72533","(UMI)AAI9305491"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider three separate topics in nonlinear optimization, one theoretical topic and two algorithmic topics. Each of these topics deals with a broad class of nonlinear optimization problems. We first introduce and analyze a generalized parametric variational inequality problem $PVI(E, T, C, \\psi$, Z) in locally convex Hausdorff topological vector spaces. Based on Nikaido's coincidence theorem, we establish several general existence theorems, even without requiring convexity nor contractibility on T and C, but merely a certain acyclic property. The case where C is not compact is considered. We also analyze asymptotic convergence of the partial proximal point algorithm for solving the generalized nonlinear equation $0\\in T(x),$ where $T : H\\to H$ is a maximal monotone multifunction, and $H := H\\sb1 \\times H\\sb2$ is a product of two real Hilbert spaces. Under the mild feasibility assumption $O\\in int(coR(T)),$ we show that the partial proximal point algorithm has the same convergence properties as does Rockafellar's proximal point algorithm. Moreover, the partial convergence rates are shown to depend upon how rapidly $T\\sp{-1}$ grows away from the solution set $T\\sb{-1}(0).$ The partial rate of convergence is linear, superlinear, or quadratic, depending upon certain parameters. Finally, we describe a primal-dual path-following interior point algorithm for solving the pair of primal-dual problems (P) and (D):","(P) sk20inf$\\sb{x}\\{f(x); Ax=b, x\\ge 0\\}$","(D) sk20sup$\\sb{x,s}\\{g(x,s); Ax=b, x\\ge 0$ and $s=\\nabla f(x) - A\\sp{T}y\\ge 0$ for some $y\\in R\\sp{m}\\}.$","Here, f is a continuously differentiable convex function in $R\\sp{n},$ and g is defined by $g(x, s) := f(x) - x\\sp{T}s.$ Under a kind of strict feasibility assumption, we show that the algorithm under modification requires a total of $O(\\sqrt{n\\ell})$ number of iterations, with total arithmetic operations of order $O(n\\sp3\\ell),$ where $\\ell$ is the initial input size. As an application to usual linear or convex quadratic programming, this algorithm solves the pair (P) and (D) in at most $O(\\sqrt{nL})$ iterations, with total arithmetic operations of order $O(n\\sp3 L),$ where L is the input size. Moreover, we show the duality gap sequence goes to zero linearly, superlinearly, or quadratically, depending upon a certain parameter $\\theta.$ Also, we show that any limit point of the induced sequence ($x\\sp{k},s\\sp{k})$ is a maximal complementary solution of a certain monotone multifunction.","Made available in DSpace on 2014-12-17T23:17:44Z (GMT). No. of bitstreams: 1 9305491.pdf: 4388713 bytes, checksum: d6c628c73ab59c4c3e6c8e38a5dbc72f (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72701 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","169 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."]},{"key":"dc:title","label":"Title","values":["Theory and Algorithms for Nonlinear Optimization and Variational Inequalities"]}]}],"canonical_facts":{"dc:contributor":["McLinden, L.,"],"dc:creator":["Chu, Liang-Ju"],"dc:date":["2014-12-17T23:17:44Z","10000-01-01","1992"],"dc:description":["We consider three separate topics in nonlinear optimization, one theoretical topic and two algorithmic topics. Each of these topics deals with a broad class of nonlinear optimization problems. We first introduce and analyze a generalized parametric variational inequality problem $PVI(E, T, C, \\psi$, Z) in locally convex Hausdorff topological vector spaces. Based on Nikaido's coincidence theorem, we establish several general existence theorems, even without requiring convexity nor contractibility on T and C, but merely a certain acyclic property. The case where C is not compact is considered. We also analyze asymptotic convergence of the partial proximal point algorithm for solving the generalized nonlinear equation $0\\in T(x),$ where $T : H\\to H$ is a maximal monotone multifunction, and $H := H\\sb1 \\times H\\sb2$ is a product of two real Hilbert spaces. Under the mild feasibility assumption $O\\in int(coR(T)),$ we show that the partial proximal point algorithm has the same convergence properties as does Rockafellar's proximal point algorithm. Moreover, the partial convergence rates are shown to depend upon how rapidly $T\\sp{-1}$ grows away from the solution set $T\\sb{-1}(0).$ The partial rate of convergence is linear, superlinear, or quadratic, depending upon certain parameters. Finally, we describe a primal-dual path-following interior point algorithm for solving the pair of primal-dual problems (P) and (D):","(P) sk20inf$\\sb{x}\\{f(x); Ax=b, x\\ge 0\\}$","(D) sk20sup$\\sb{x,s}\\{g(x,s); Ax=b, x\\ge 0$ and $s=\\nabla f(x) - A\\sp{T}y\\ge 0$ for some $y\\in R\\sp{m}\\}.$","Here, f is a continuously differentiable convex function in $R\\sp{n},$ and g is defined by $g(x, s) := f(x) - x\\sp{T}s.$ Under a kind of strict feasibility assumption, we show that the algorithm under modification requires a total of $O(\\sqrt{n\\ell})$ number of iterations, with total arithmetic operations of order $O(n\\sp3\\ell),$ where $\\ell$ is the initial input size. As an application to usual linear or convex quadratic programming, this algorithm solves the pair (P) and (D) in at most $O(\\sqrt{nL})$ iterations, with total arithmetic operations of order $O(n\\sp3 L),$ where L is the input size. Moreover, we show the duality gap sequence goes to zero linearly, superlinearly, or quadratically, depending upon a certain parameter $\\theta.$ Also, we show that any limit point of the induced sequence ($x\\sp{k},s\\sp{k})$ is a maximal complementary solution of a certain monotone multifunction.","Made available in DSpace on 2014-12-17T23:17:44Z (GMT). No. of bitstreams: 1 9305491.pdf: 4388713 bytes, checksum: d6c628c73ab59c4c3e6c8e38a5dbc72f (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72701 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","169 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."],"dc:identifier":["http://hdl.handle.net/2142/72533","(UMI)AAI9305491"],"dc:subject":["Mathematics"],"dc:title":["Theory and Algorithms for Nonlinear Optimization and Variational Inequalities"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}