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Jean-Jacques Slotine. Soto, Jonathan Massachusetts Institute of Technology. Dept. of Mechanical Engineering. Massachusetts Institute of Technology. Department of Mechanical Engineering 2011-04-25T16:15:42Z 2011-04-25T16:15:42Z 2010 2010 http://hdl.handle.net/1721.1/62538 712952313 Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010. Cataloged from PDF version of thesis. Includes bibliographical references (p. 77-78). This thesis derives new results linking nonlinear contraction analysis, a recent stability theory for nonlinear systems, and constrained optimization theory. Although dynamic systems and optimization are both areas that have been extensively studied [21], few results have been achieved in this direction because strong enough tools for dynamic systems were not available. Contraction analysis provides the necessary mathematical background. Based on an operator that projects the speed of the system on the tangent space of the constraints, we derive generalizations of Lagrange parameters. After presenting some initial examples that show the relations between contraction and optimization, we derive a contraction theorem for nonlinear systems with equality constraints. The method is applied to examples in differential geometry and biological systems. A new physical interpretation of Lagrange parameters is provided. In the autonomous case, we derive a new algorithm to solve minimization problems. Next, we state a contraction theorem for nonlinear systems with inequality constraints. In the autonomous case, the algorithm solves minimization problems very fast compared to standard algorithms. Finally, we state another contraction theorem for nonlinear systems with time-varying equality constraints. A new generalization of time varying Lagrange parameters is given. In the autonomous case, we provide a solution for a new class of optimization problems, minimization with time-varying constraints. by Jonathan Soto. S.M. 78 p. eng Massachusetts Institute of Technology M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 Mechanical Engineering. Nonlinear contraction tools for constrained optimization Thesis application/pdf Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="326ccadc-b0b7-49e9-842c-5df7ebdca069"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>eng</Language> <Title>Nonlinear contraction tools for constrained optimization</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2010</PublicationDate> <Authors> <Author> <DisplayName>Soto, Jonathan</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://dspace.mit.edu/handle/1721.1/7582</License> <Keyword>Mechanical Engineering.</Keyword> <Abstract>This thesis derives new results linking nonlinear contraction analysis, a recent stability theory for nonlinear systems, and constrained optimization theory. Although dynamic systems and optimization are both areas that have been extensively studied [21], few results have been achieved in this direction because strong enough tools for dynamic systems were not available. Contraction analysis provides the necessary mathematical background. Based on an operator that projects the speed of the system on the tangent space of the constraints, we derive generalizations of Lagrange parameters. After presenting some initial examples that show the relations between contraction and optimization, we derive a contraction theorem for nonlinear systems with equality constraints. The method is applied to examples in differential geometry and biological systems. A new physical interpretation of Lagrange parameters is provided. In the autonomous case, we derive a new algorithm to solve minimization problems. Next, we state a contraction theorem for nonlinear systems with inequality constraints. In the autonomous case, the algorithm solves minimization problems very fast compared to standard algorithms. Finally, we state another contraction theorem for nonlinear systems with time-varying equality constraints. A new generalization of time varying Lagrange parameters is given. In the autonomous case, we provide a solution for a new class of optimization problems, minimization with time-varying constraints.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>