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Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen
| Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen |
| Autore | Lauritzen Niels <1964-> |
| Pubbl/distr/stampa | Singapore, : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (300 p.) |
| Disciplina | 515.88 |
| Soggetto topico |
Convex domains
Algebras, linear |
| Soggetto genere / forma | Electronic books. |
| ISBN | 981-4412-52-X |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; Acknowledgments; Contents; 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; Minkowski sum, dilation and the polar of a subset; 3.2 The convex hull; 3.3 Faces of convex subsets; Interlude: Integral points in convex subsets; 3.4 Convex cones; The recession cone; Finitely generated cones
3.5 Carathéodory's theorem3.6 The convex hull, simplicial subsets and Bland's rule; Non-cycling; 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; 4.5.1 Markov chains and steady states; 4.5.2 Gordan's theorem; 4.5.3 Duality in linear programming; 4.6 Minkowski's theorem; 4.7 Parametrization of polyhedra; 4.8 Doubly stochastic matrices: The Birkhoff polytope; 4.8.1 Perfect pairings and doubly stochastic matrices; 4.9 Exercises; 5. Computations with polyhedra 5.1 Extreme rays and minimal generators in convex cones5.2 Minimal generators of a polyhedral cone; 5.3 The double description method; 5.3.1 Converting from half space to vertex representation; 5.3.2 Converting from vertex to half space representation; 5.3.3 Computing the convex hull; 5.4 Linear programming and the simplex algorithm; 5.4.1 Two examples of linear programs; 5.4.2 The simplex algorithm in a special case; 5.4.3 The simplex algorithm for polyhedra in general form; 5.4.4 The simplicial hack; 5.4.5 The computational miracle of the simplex tableau; The simplex algorithm Explaining the steps5.4.6 Computing a vertex in a polyhedron; 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; 7.5.1 The Newton-Raphson method for finding roots; 7.5.2 Critical points and extrema; 7.6 Taylor polynomials; 7.7 Differentiable convex functions; 7.8 Exercises 8. Differentiable functions of several variables8.1 Differentiability; 8.1.1 The Newton-Raphson method for several variables; 8.1.2 Local extrema for functions of several variables; 8.2 The chain rule; 8.3 Lagrange multipliers; The two variable case; The general case and the Lagrangian; 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 |
| Record Nr. | UNINA-9910462802103321 |
Lauritzen Niels <1964->
|
||
| Singapore, : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen
| Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen |
| Autore | Lauritzen Niels <1964-> |
| Pubbl/distr/stampa | Singapore, : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (300 p.) |
| Disciplina | 515.88 |
| Soggetto topico |
Convex domains
Algebras, linear |
| ISBN | 981-4412-52-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; Acknowledgments; Contents; 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; Minkowski sum, dilation and the polar of a subset; 3.2 The convex hull; 3.3 Faces of convex subsets; Interlude: Integral points in convex subsets; 3.4 Convex cones; The recession cone; Finitely generated cones
3.5 Carathéodory's theorem3.6 The convex hull, simplicial subsets and Bland's rule; Non-cycling; 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; 4.5.1 Markov chains and steady states; 4.5.2 Gordan's theorem; 4.5.3 Duality in linear programming; 4.6 Minkowski's theorem; 4.7 Parametrization of polyhedra; 4.8 Doubly stochastic matrices: The Birkhoff polytope; 4.8.1 Perfect pairings and doubly stochastic matrices; 4.9 Exercises; 5. Computations with polyhedra 5.1 Extreme rays and minimal generators in convex cones5.2 Minimal generators of a polyhedral cone; 5.3 The double description method; 5.3.1 Converting from half space to vertex representation; 5.3.2 Converting from vertex to half space representation; 5.3.3 Computing the convex hull; 5.4 Linear programming and the simplex algorithm; 5.4.1 Two examples of linear programs; 5.4.2 The simplex algorithm in a special case; 5.4.3 The simplex algorithm for polyhedra in general form; 5.4.4 The simplicial hack; 5.4.5 The computational miracle of the simplex tableau; The simplex algorithm Explaining the steps5.4.6 Computing a vertex in a polyhedron; 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; 7.5.1 The Newton-Raphson method for finding roots; 7.5.2 Critical points and extrema; 7.6 Taylor polynomials; 7.7 Differentiable convex functions; 7.8 Exercises 8. Differentiable functions of several variables8.1 Differentiability; 8.1.1 The Newton-Raphson method for several variables; 8.1.2 Local extrema for functions of several variables; 8.2 The chain rule; 8.3 Lagrange multipliers; The two variable case; The general case and the Lagrangian; 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 |
| Record Nr. | UNINA-9910786967903321 |
|
Lauritzen Niels <1964->
|
||
| Singapore, : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen
| Undergraduate convexity [[electronic resource] ] : from Fourier and Motzkin to Kuhn and Tucker / / Niels Lauritzen |
| Autore | Lauritzen Niels <1964-> |
| Pubbl/distr/stampa | Singapore, : World Scientific, 2013 |
| Descrizione fisica | 1 online resource (300 p.) |
| Disciplina | 515.88 |
| Soggetto topico |
Convex domains
Algebras, linear |
| ISBN | 981-4412-52-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Preface; Acknowledgments; Contents; 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; Minkowski sum, dilation and the polar of a subset; 3.2 The convex hull; 3.3 Faces of convex subsets; Interlude: Integral points in convex subsets; 3.4 Convex cones; The recession cone; Finitely generated cones
3.5 Carathéodory's theorem3.6 The convex hull, simplicial subsets and Bland's rule; Non-cycling; 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; 4.5.1 Markov chains and steady states; 4.5.2 Gordan's theorem; 4.5.3 Duality in linear programming; 4.6 Minkowski's theorem; 4.7 Parametrization of polyhedra; 4.8 Doubly stochastic matrices: The Birkhoff polytope; 4.8.1 Perfect pairings and doubly stochastic matrices; 4.9 Exercises; 5. Computations with polyhedra 5.1 Extreme rays and minimal generators in convex cones5.2 Minimal generators of a polyhedral cone; 5.3 The double description method; 5.3.1 Converting from half space to vertex representation; 5.3.2 Converting from vertex to half space representation; 5.3.3 Computing the convex hull; 5.4 Linear programming and the simplex algorithm; 5.4.1 Two examples of linear programs; 5.4.2 The simplex algorithm in a special case; 5.4.3 The simplex algorithm for polyhedra in general form; 5.4.4 The simplicial hack; 5.4.5 The computational miracle of the simplex tableau; The simplex algorithm Explaining the steps5.4.6 Computing a vertex in a polyhedron; 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; 7.5.1 The Newton-Raphson method for finding roots; 7.5.2 Critical points and extrema; 7.6 Taylor polynomials; 7.7 Differentiable convex functions; 7.8 Exercises 8. Differentiable functions of several variables8.1 Differentiability; 8.1.1 The Newton-Raphson method for several variables; 8.1.2 Local extrema for functions of several variables; 8.2 The chain rule; 8.3 Lagrange multipliers; The two variable case; The general case and the Lagrangian; 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 |
| Record Nr. | UNINA-9910820519603321 |
|
Lauritzen Niels <1964->
|
||
| Singapore, : World Scientific, 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||