Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
Conditions of Use
This book is licensed under a Creative Commons License (CC BY-NC-SA). You can download the ebook Convex Optimization for free.
- Title
- Convex Optimization
- Publisher
- Cambridge University Press
- Author(s)
- Lieven Vandenberghe, Stephen Boyd
- Published
- 2004-03-25
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 727
- Language
- English
- ISBN-10
- 0521833787
- ISBN-13
- 9780521833783
- License
- CC BY-NC-SA
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Preface Introduction Mathematical optimization Least-squares and linear programming Convex optimization Nonlinear optimization Outline Notation Bibliography I Theory Convex sets Affine and convex sets Some important examples Operations that preserve convexity Generalized inequalities Separating and supporting hyperplanes Dual cones and generalized inequalities Bibliography Exercises Convex functions Basic properties and examples Operations that preserve convexity The conjugate function Quasiconvex functions Log-concave and log-convex functions Convexity with respect to generalized inequalities Bibliography Exercises Convex optimization problems Optimization problems Convex optimization Linear optimization problems Quadratic optimization problems Geometric programming Generalized inequality constraints Vector optimization Bibliography Exercises Duality The Lagrange dual function The Lagrange dual problem Geometric interpretation Saddle-point interpretation Optimality conditions Perturbation and sensitivity analysis Examples Theorems of alternatives Generalized inequalities Bibliography Exercises II Applications Approximation and fitting Norm approximation Least-norm problems Regularized approximation Robust approximation Function fitting and interpolation Bibliography Exercises Statistical estimation Parametric distribution estimation Nonparametric distribution estimation Optimal detector design and hypothesis testing Chebyshev and Chernoff bounds Experiment design Bibliography Exercises Geometric problems Projection on a set Distance between sets Euclidean distance and angle problems Extremal volume ellipsoids Centering Classification Placement and location Floor planning Bibliography Exercises III Algorithms Unconstrained minimization Unconstrained minimization problems Descent methods Gradient descent method Steepest descent method Newton's method Self-concordance Implementation Bibliography Exercises Equality constrained minimization Equality constrained minimization problems Newton's method with equality constraints Infeasible start Newton method Implementation Bibliography Exercises Interior-point methods Inequality constrained minimization problems Logarithmic barrier function and central path The barrier method Feasibility and phase I methods Complexity analysis via self-concordance Problems with generalized inequalities Primal-dual interior-point methods Implementation Bibliography Exercises Appendices Mathematical background Norms Analysis Functions Derivatives Linear algebra Bibliography Problems involving two quadratic functions Single constraint quadratic optimization The S-procedure The field of values of two symmetric matrices Proofs of the strong duality results Bibliography Numerical linear algebra background Matrix structure and algorithm complexity Solving linear equations with factored matrices LU, Cholesky, and LDLT factorization Block elimination and Schur complements Solving underdetermined linear equations Bibliography References Notation
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