Undergraduate convexity: from Fourier and Motzkin to Kuhn and Tucker
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
[Hackensack] New Jersey
World Scientific
[2013]
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| Subjects: | |
| Online Access: | FAW01 FAW02 Volltext |
| Item Description: | Description based on print version record |
| Physical Description: | 1 online resource (xiv, 283 pages) illustrations |
| ISBN: | 1299556337 9781299556331 9789814412513 9789814412520 9789814412537 9789814452762 9814412511 981441252X 9814412538 9814452769 |
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MARC
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| 100 | 1 | |a Lauritzen, Niels |d 1964- |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Undergraduate convexity |b from Fourier and Motzkin to Kuhn and Tucker |c Niels Lauritzen |
| 264 | 1 | |a [Hackensack] New Jersey |b World Scientific |c [2013] | |
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| 505 | 8 | |a Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples. Starting from linear inequalities and Fourier-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point algorithm--Page [4] of cover | |
| 505 | 8 | |a 1. Fourier-Motzkin elimination. 1.1. Linear inequalities. 1.2. Linear optimization using elimination. 1.3. Polyhedra. 1.4. Exercises -- 2. Affine subspaces. 2.1. Definition and basics. 2.2. The affine hull. 2.3. Affine subspaces and subspaces. 2.4. Affine independence and the dimension of a subset. 2.5. Exercises -- 3. Convex subsets. 3.1. Basics. 3.2. The convex hull. 3.3. Faces of convex subsets. 3.4. Convex cones. 3.5. Carathéodory's theorem. 3.6. The convex hull, simplicial subsets and Bland's rule. 3.7. Exercises -- 4. Polyhedra. 4.1. Faces of polyhedra. 4.2. Extreme points and linear optimization. 4.3. Weyl's theorem. 4.4. Farkas's lemma. 4.5. Three applications of Farkas's lemma. 4.6. Minkowski's theorem. 4.7. Parametrization of polyhedra. 4.8. Doubly stochastic matrices: the Birkhoff polytope. 4.9. Exercises -- | |
| 505 | 8 | |a 5. Computations with polyhedra. 5.1. Extreme rays and minimal generators in convex cones. 5.2. Minimal generators of a polyhedral cone. 5.3. The double description method. 5.4. Linear programming and the simplex algorithm. 5.5. Exercises -- 6. Closed convex subsets and separating hyperplanes. 6.1. Closed convex subsets. 6.2. Supporting hyperplanes. 6.3. Separation by hyperplanes. 6.4. Exercises. 7. Convex functions. 7.1. Basics. 7.2. Jensen's inequality. 7.3. Minima of convex functions. 7.4. Convex functions of one variable. 7.5. Differentiable functions of one variable. 7.6. Taylor polynomials. 7.7. Differentiable convex functions. 7.8. Exercises -- 8. Differentiable functions of several variables. 8.1. Differentiability. 8.2. The chain rule. 8.3. Lagrange multipliers. 8.4. The arithmetic-geometric inequality revisited. 8.5. Exercises -- | |
| 505 | 8 | |a 9. Convex functions of several variables. 9.1. Subgradients. 9.2. Convexity and the Hessian. 9.3. Positive definite and positive semidefinite matrices. 9.4. Principal minors and definite matrices. 9.5. The positive semidefinite cone. 9.6. Reduction of symmetric matrices. 9.7. The spectral theorem. 9.8. Quadratic forms. 9.9. Exercises -- 10. Convex optimization. 10.1. A geometric optimality criterion. 10.2. The Karush-Kuhn-Tucker conditions. 10.3. An example. 10.4. The Langrangian, saddle points, duality and game theory. 10.5. An interior point method. 10.6. Maximizing convex functions over polytopes. 10.7. Exercises | |
| 650 | 7 | |a MATHEMATICS / Calculus |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS / Mathematical Analysis |2 bisacsh | |
| 650 | 4 | |a Convex functions | |
| 650 | 4 | |a Convex domains | |
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Record in the Search Index
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| any_adam_object | |
| author | Lauritzen, Niels 1964- |
| author_facet | Lauritzen, Niels 1964- |
| author_role | aut |
| author_sort | Lauritzen, Niels 1964- |
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| bvnumber | BV043036547 |
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| contents | Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples. Starting from linear inequalities and Fourier-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point algorithm--Page [4] of cover 1. Fourier-Motzkin elimination. 1.1. Linear inequalities. 1.2. Linear optimization using elimination. 1.3. Polyhedra. 1.4. Exercises -- 2. Affine subspaces. 2.1. Definition and basics. 2.2. The affine hull. 2.3. Affine subspaces and subspaces. 2.4. Affine independence and the dimension of a subset. 2.5. Exercises -- 3. Convex subsets. 3.1. Basics. 3.2. The convex hull. 3.3. Faces of convex subsets. 3.4. Convex cones. 3.5. Carathéodory's theorem. 3.6. The convex hull, simplicial subsets and Bland's rule. 3.7. Exercises -- 4. Polyhedra. 4.1. Faces of polyhedra. 4.2. Extreme points and linear optimization. 4.3. Weyl's theorem. 4.4. Farkas's lemma. 4.5. Three applications of Farkas's lemma. 4.6. Minkowski's theorem. 4.7. Parametrization of polyhedra. 4.8. Doubly stochastic matrices: the Birkhoff polytope. 4.9. Exercises -- 5. Computations with polyhedra. 5.1. Extreme rays and minimal generators in convex cones. 5.2. Minimal generators of a polyhedral cone. 5.3. The double description method. 5.4. Linear programming and the simplex algorithm. 5.5. Exercises -- 6. Closed convex subsets and separating hyperplanes. 6.1. Closed convex subsets. 6.2. Supporting hyperplanes. 6.3. Separation by hyperplanes. 6.4. Exercises. 7. Convex functions. 7.1. Basics. 7.2. Jensen's inequality. 7.3. Minima of convex functions. 7.4. Convex functions of one variable. 7.5. Differentiable functions of one variable. 7.6. Taylor polynomials. 7.7. Differentiable convex functions. 7.8. Exercises -- 8. Differentiable functions of several variables. 8.1. Differentiability. 8.2. The chain rule. 8.3. Lagrange multipliers. 8.4. The arithmetic-geometric inequality revisited. 8.5. Exercises -- 9. Convex functions of several variables. 9.1. Subgradients. 9.2. Convexity and the Hessian. 9.3. Positive definite and positive semidefinite matrices. 9.4. Principal minors and definite matrices. 9.5. The positive semidefinite cone. 9.6. Reduction of symmetric matrices. 9.7. The spectral theorem. 9.8. Quadratic forms. 9.9. Exercises -- 10. Convex optimization. 10.1. A geometric optimality criterion. 10.2. The Karush-Kuhn-Tucker conditions. 10.3. An example. 10.4. The Langrangian, saddle points, duality and game theory. 10.5. An interior point method. 10.6. Maximizing convex functions over polytopes. 10.7. Exercises |
| ctrlnum | (OCoLC)843871633 (DE-599)BVBBV043036547 |
| dewey-full | 515.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.8 |
| dewey-search | 515.8 |
| dewey-sort | 3515.8 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| indexdate | 2024-07-10T07:15:38Z |
| institution | BVB |
| isbn | 1299556337 9781299556331 9789814412513 9789814412520 9789814412537 9789814452762 9814412511 981441252X 9814412538 9814452769 |
| language | English |
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| publisher | World Scientific |
| record_format | marc |
| spelling | Lauritzen, Niels 1964- Verfasser aut Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker Niels Lauritzen [Hackensack] New Jersey World Scientific [2013] © 2013 1 online resource (xiv, 283 pages) illustrations txt rdacontent c rdamedia cr rdacarrier Description based on print version record Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples. Starting from linear inequalities and Fourier-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point algorithm--Page [4] of cover 1. Fourier-Motzkin elimination. 1.1. Linear inequalities. 1.2. Linear optimization using elimination. 1.3. Polyhedra. 1.4. Exercises -- 2. Affine subspaces. 2.1. Definition and basics. 2.2. The affine hull. 2.3. Affine subspaces and subspaces. 2.4. Affine independence and the dimension of a subset. 2.5. Exercises -- 3. Convex subsets. 3.1. Basics. 3.2. The convex hull. 3.3. Faces of convex subsets. 3.4. Convex cones. 3.5. Carathéodory's theorem. 3.6. The convex hull, simplicial subsets and Bland's rule. 3.7. Exercises -- 4. Polyhedra. 4.1. Faces of polyhedra. 4.2. Extreme points and linear optimization. 4.3. Weyl's theorem. 4.4. Farkas's lemma. 4.5. Three applications of Farkas's lemma. 4.6. Minkowski's theorem. 4.7. Parametrization of polyhedra. 4.8. Doubly stochastic matrices: the Birkhoff polytope. 4.9. Exercises -- 5. Computations with polyhedra. 5.1. Extreme rays and minimal generators in convex cones. 5.2. Minimal generators of a polyhedral cone. 5.3. The double description method. 5.4. Linear programming and the simplex algorithm. 5.5. Exercises -- 6. Closed convex subsets and separating hyperplanes. 6.1. Closed convex subsets. 6.2. Supporting hyperplanes. 6.3. Separation by hyperplanes. 6.4. Exercises. 7. Convex functions. 7.1. Basics. 7.2. Jensen's inequality. 7.3. Minima of convex functions. 7.4. Convex functions of one variable. 7.5. Differentiable functions of one variable. 7.6. Taylor polynomials. 7.7. Differentiable convex functions. 7.8. Exercises -- 8. Differentiable functions of several variables. 8.1. Differentiability. 8.2. The chain rule. 8.3. Lagrange multipliers. 8.4. The arithmetic-geometric inequality revisited. 8.5. Exercises -- 9. Convex functions of several variables. 9.1. Subgradients. 9.2. Convexity and the Hessian. 9.3. Positive definite and positive semidefinite matrices. 9.4. Principal minors and definite matrices. 9.5. The positive semidefinite cone. 9.6. Reduction of symmetric matrices. 9.7. The spectral theorem. 9.8. Quadratic forms. 9.9. Exercises -- 10. Convex optimization. 10.1. A geometric optimality criterion. 10.2. The Karush-Kuhn-Tucker conditions. 10.3. An example. 10.4. The Langrangian, saddle points, duality and game theory. 10.5. An interior point method. 10.6. Maximizing convex functions over polytopes. 10.7. Exercises MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh Convex functions Convex domains Konvexe Funktion (DE-588)4139679-0 gnd rswk-swf Konvexität (DE-588)4114284-6 gnd rswk-swf Konvexe Menge (DE-588)4165212-5 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Konvexität (DE-588)4114284-6 s Konvexe Menge (DE-588)4165212-5 s Konvexe Funktion (DE-588)4139679-0 s 2\p DE-604 Erscheint auch als Druck-Ausgabe Lauritzen, Niels, 1964- Undergraduate convexity http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=575386 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lauritzen, Niels 1964- Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples. Starting from linear inequalities and Fourier-Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush-Kuhn-Tucker conditions, duality and an interior point algorithm--Page [4] of cover 1. Fourier-Motzkin elimination. 1.1. Linear inequalities. 1.2. Linear optimization using elimination. 1.3. Polyhedra. 1.4. Exercises -- 2. Affine subspaces. 2.1. Definition and basics. 2.2. The affine hull. 2.3. Affine subspaces and subspaces. 2.4. Affine independence and the dimension of a subset. 2.5. Exercises -- 3. Convex subsets. 3.1. Basics. 3.2. The convex hull. 3.3. Faces of convex subsets. 3.4. Convex cones. 3.5. Carathéodory's theorem. 3.6. The convex hull, simplicial subsets and Bland's rule. 3.7. Exercises -- 4. Polyhedra. 4.1. Faces of polyhedra. 4.2. Extreme points and linear optimization. 4.3. Weyl's theorem. 4.4. Farkas's lemma. 4.5. Three applications of Farkas's lemma. 4.6. Minkowski's theorem. 4.7. Parametrization of polyhedra. 4.8. Doubly stochastic matrices: the Birkhoff polytope. 4.9. Exercises -- 5. Computations with polyhedra. 5.1. Extreme rays and minimal generators in convex cones. 5.2. Minimal generators of a polyhedral cone. 5.3. The double description method. 5.4. Linear programming and the simplex algorithm. 5.5. Exercises -- 6. Closed convex subsets and separating hyperplanes. 6.1. Closed convex subsets. 6.2. Supporting hyperplanes. 6.3. Separation by hyperplanes. 6.4. Exercises. 7. Convex functions. 7.1. Basics. 7.2. Jensen's inequality. 7.3. Minima of convex functions. 7.4. Convex functions of one variable. 7.5. Differentiable functions of one variable. 7.6. Taylor polynomials. 7.7. Differentiable convex functions. 7.8. Exercises -- 8. Differentiable functions of several variables. 8.1. Differentiability. 8.2. The chain rule. 8.3. Lagrange multipliers. 8.4. The arithmetic-geometric inequality revisited. 8.5. Exercises -- 9. Convex functions of several variables. 9.1. Subgradients. 9.2. Convexity and the Hessian. 9.3. Positive definite and positive semidefinite matrices. 9.4. Principal minors and definite matrices. 9.5. The positive semidefinite cone. 9.6. Reduction of symmetric matrices. 9.7. The spectral theorem. 9.8. Quadratic forms. 9.9. Exercises -- 10. Convex optimization. 10.1. A geometric optimality criterion. 10.2. The Karush-Kuhn-Tucker conditions. 10.3. An example. 10.4. The Langrangian, saddle points, duality and game theory. 10.5. An interior point method. 10.6. Maximizing convex functions over polytopes. 10.7. Exercises MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh Convex functions Convex domains Konvexe Funktion (DE-588)4139679-0 gnd Konvexität (DE-588)4114284-6 gnd Konvexe Menge (DE-588)4165212-5 gnd |
| subject_GND | (DE-588)4139679-0 (DE-588)4114284-6 (DE-588)4165212-5 (DE-588)4123623-3 |
| title | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker |
| title_auth | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker |
| title_exact_search | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker |
| title_full | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker Niels Lauritzen |
| title_fullStr | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker Niels Lauritzen |
| title_full_unstemmed | Undergraduate convexity from Fourier and Motzkin to Kuhn and Tucker Niels Lauritzen |
| title_short | Undergraduate convexity |
| title_sort | undergraduate convexity from fourier and motzkin to kuhn and tucker |
| title_sub | from Fourier and Motzkin to Kuhn and Tucker |
| topic | MATHEMATICS / Calculus bisacsh MATHEMATICS / Mathematical Analysis bisacsh Convex functions Convex domains Konvexe Funktion (DE-588)4139679-0 gnd Konvexität (DE-588)4114284-6 gnd Konvexe Menge (DE-588)4165212-5 gnd |
| topic_facet | MATHEMATICS / Calculus MATHEMATICS / Mathematical Analysis Convex functions Convex domains Konvexe Funktion Konvexität Konvexe Menge Lehrbuch |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=575386 |
| work_keys_str_mv | AT lauritzenniels undergraduateconvexityfromfourierandmotzkintokuhnandtucker |