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作者 Lauritzen, Niels, 1964-
書名 Undergraduate convexity : from fourier and motzkin to kuhn and tucker / Niels Lauritzen,
出版項 Singapore : World Scientific, c2013
國際標準書號 9789814412513
9789814412537 (e-bk)
9814412538 (e-bk)
book jacket
館藏地 索書號 處理狀態 OPAC 訊息 條碼
 數學所圖書室  QA639.5 L386 2013    在架上    30340200529448
說明 283 p. : ill
附註 Bibliogr.: p. 273-275
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
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
主題 Géométrie convexe
Algèbres convexes
Fonctions convexes
Ensembles convexes
Mathématiques -- Manuels d'enseignement supérieur
Alt Author Lauritzen, Niels, 1964-
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