Nonlinear Optimization in Finite Dimensions: Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Springer US
2001
|
| Schriftenreihe: | Nonconvex Optimization and Its Applications
47 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization |
| Beschreibung: | 1 Online-Ressource (X, 510 p) |
| ISBN: | 9781461500179 9781461348870 |
| ISSN: | 1571-568X |
| DOI: | 10.1007/978-1-4615-0017-9 |
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| 500 | |a At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization | ||
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| institution | BVB |
| isbn | 9781461500179 9781461348870 |
| issn | 1571-568X |
| language | English |
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| spelling | Jongen, Hubertus Th Verfasser aut Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects by Hubertus Th. Jongen, Peter Jonker, Frank Twilt Boston, MA Springer US 2001 1 Online-Ressource (X, 510 p) txt rdacontent c rdamedia cr rdacarrier Nonconvex Optimization and Its Applications 47 1571-568X At the heart of the topology of global optimization lies Morse Theory: The study of the behaviour of lower level sets of functions as the level varies. Roughly speaking, the topology of lower level sets only may change when passing a level which corresponds to a stationary point (or Karush-Kuhn Tucker point). We study elements of Morse Theory, both in the unconstrained and constrained case. Special attention is paid to the degree of differentiabil ity of the functions under consideration. The reader will become motivated to discuss the possible shapes and forms of functions that may possibly arise within a given problem framework. In a separate chapter we show how certain ideas may be carried over to nonsmooth items, such as problems of Chebyshev approximation type. We made this choice in order to show that a good under standing of regular smooth problems may lead to a straightforward treatment of "just" continuous problems by means of suitable perturbation techniques, taking a priori nonsmoothness into account. Moreover, we make a focal point analysis in order to emphasize the difference between inner product norms and, for example, the maximum norm. Then, specific tools from algebraic topol ogy, in particular homology theory, are treated in some detail. However, this development is carried out only as far as it is needed to understand the relation between critical points of a function on a manifold with structured boundary. Then, we pay attention to three important subjects in nonlinear optimization Mathematics Global analysis Differential Equations Mathematical optimization Algebraic topology Optimization Global Analysis and Analysis on Manifolds Calculus of Variations and Optimal Control; Optimization Ordinary Differential Equations Algebraic Topology Mathematik Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 s 1\p DE-604 Jonker, Peter Sonstige oth Twilt, Frank Sonstige oth https://doi.org/10.1007/978-1-4615-0017-9 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Jongen, Hubertus Th Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects Mathematics Global analysis Differential Equations Mathematical optimization Algebraic topology Optimization Global Analysis and Analysis on Manifolds Calculus of Variations and Optimal Control; Optimization Ordinary Differential Equations Algebraic Topology Mathematik Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| subject_GND | (DE-588)4128192-5 |
| title | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects |
| title_auth | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects |
| title_exact_search | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects |
| title_full | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects by Hubertus Th. Jongen, Peter Jonker, Frank Twilt |
| title_fullStr | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects by Hubertus Th. Jongen, Peter Jonker, Frank Twilt |
| title_full_unstemmed | Nonlinear Optimization in Finite Dimensions Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects by Hubertus Th. Jongen, Peter Jonker, Frank Twilt |
| title_short | Nonlinear Optimization in Finite Dimensions |
| title_sort | nonlinear optimization in finite dimensions morse theory chebyshev approximation transversality flows parametric aspects |
| title_sub | Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects |
| topic | Mathematics Global analysis Differential Equations Mathematical optimization Algebraic topology Optimization Global Analysis and Analysis on Manifolds Calculus of Variations and Optimal Control; Optimization Ordinary Differential Equations Algebraic Topology Mathematik Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| topic_facet | Mathematics Global analysis Differential Equations Mathematical optimization Algebraic topology Optimization Global Analysis and Analysis on Manifolds Calculus of Variations and Optimal Control; Optimization Ordinary Differential Equations Algebraic Topology Mathematik Nichtlineare Optimierung |
| url | https://doi.org/10.1007/978-1-4615-0017-9 |
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