Abstract Convexity and Global Optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Boston, MA
Springer US
2000
|
| Series: | Nonconvex Optimization and Its Applications
44 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to accomplish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actually plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neighbourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support function. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support |
| Physical Description: | 1 Online-Ressource (XVIII, 493 p) |
| ISBN: | 9781475732009 9781441948311 |
| ISSN: | 1571-568X |
| DOI: | 10.1007/978-1-4757-3200-9 |
Staff View
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| 500 | |a Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to accomplish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actually plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neighbourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support function. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support | ||
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| id | DE-604.BV042421437 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:09Z |
| institution | BVB |
| isbn | 9781475732009 9781441948311 |
| issn | 1571-568X |
| language | English |
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| physical | 1 Online-Ressource (XVIII, 493 p) |
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| publishDate | 2000 |
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| publishDateSort | 2000 |
| publisher | Springer US |
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| series | Nonconvex Optimization and Its Applications |
| series2 | Nonconvex Optimization and Its Applications |
| spelling | Rubinov, Alexander Verfasser aut Abstract Convexity and Global Optimization by Alexander Rubinov Boston, MA Springer US 2000 1 Online-Ressource (XVIII, 493 p) txt rdacontent c rdamedia cr rdacarrier Nonconvex Optimization and Its Applications 44 1571-568X Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to accomplish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actually plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neighbourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support function. Since f(y) ~ h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support Mathematics Mathematical optimization Computer engineering Calculus of Variations and Optimal Control; Optimization Optimization Mathematical Modeling and Industrial Mathematics Electrical Engineering Mathematik Nonconvex Optimization and Its Applications 44 (DE-604)BV010085908 44 https://doi.org/10.1007/978-1-4757-3200-9 Verlag Volltext |
| spellingShingle | Rubinov, Alexander Abstract Convexity and Global Optimization Nonconvex Optimization and Its Applications Mathematics Mathematical optimization Computer engineering Calculus of Variations and Optimal Control; Optimization Optimization Mathematical Modeling and Industrial Mathematics Electrical Engineering Mathematik |
| title | Abstract Convexity and Global Optimization |
| title_auth | Abstract Convexity and Global Optimization |
| title_exact_search | Abstract Convexity and Global Optimization |
| title_full | Abstract Convexity and Global Optimization by Alexander Rubinov |
| title_fullStr | Abstract Convexity and Global Optimization by Alexander Rubinov |
| title_full_unstemmed | Abstract Convexity and Global Optimization by Alexander Rubinov |
| title_short | Abstract Convexity and Global Optimization |
| title_sort | abstract convexity and global optimization |
| topic | Mathematics Mathematical optimization Computer engineering Calculus of Variations and Optimal Control; Optimization Optimization Mathematical Modeling and Industrial Mathematics Electrical Engineering Mathematik |
| topic_facet | Mathematics Mathematical optimization Computer engineering Calculus of Variations and Optimal Control; Optimization Optimization Mathematical Modeling and Industrial Mathematics Electrical Engineering Mathematik |
| url | https://doi.org/10.1007/978-1-4757-3200-9 |
| volume_link | (DE-604)BV010085908 |
| work_keys_str_mv | AT rubinovalexander abstractconvexityandglobaloptimization |