Linear and nonlinear optimization 1st edition by Richard Cottle, Mukund Thapa – Ebook PDF Instant Download/Delivery: 1493970551, 9781493970551
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ISBN 10: 1493970551
ISBN 13: 9781493970551
Author: Richard Cottle, Mukund Thapa
This textbook on Linear and Nonlinear Optimization is intended for graduate and advanced undergraduate students in operations research and related fields. It is both literate and mathematically strong, yet requires no prior course in optimization. As suggested by its title, the book is divided into two parts covering in their individual chapters LP Models and Applications; Linear Equations and Inequalities; The Simplex Algorithm; Simplex Algorithm Continued; Duality and the Dual Simplex Algorithm; Postoptimality Analyses; Computational Considerations; Nonlinear (NLP) Models and Applications; Unconstrained Optimization; Descent Methods; Optimality Conditions; Problems with Linear Constraints; Problems with Nonlinear Constraints; Interior-Point Methods; and an Appendix covering Mathematical Concepts. Each chapter ends with a set of exercises. The book is based on lecture notes the authors have used in numerous optimization courses the authors have taught at StanfordUniversity. It emphasizes modeling and numerical algorithms for optimization with continuous (not integer) variables. The discussion presents the underlying theory without always focusing on formal mathematical proofs (which can be found in cited references). Another feature of this book is its inclusion of cultural and historical matters, most often appearing among the footnotes. “This book is a real gem. The authors do a masterful job of rigorously presenting all of the relevant theory clearly and concisely while managing to avoid unnecessary tedious mathematical details. This is an ideal book for teaching a one or two semester masters-level course in optimization – it broadly covers linear and nonlinear programming effectively balancing modeling, algorithmic theory, computation, implementation, illuminating historical facts, and numerous interesting examples and exercises. Due to the clarity of the exposition, this book also serves as avaluable reference for self-study.” Professor Ilan Adler, IEOR Department, UC Berkeley “A carefully crafted introduction to the main elements and applications of mathematical optimization. This volume presents the essential concepts of linear and nonlinear programming in an accessible format filled with anecdotes, examples, and exercises that bring the topic to life. The authors plumb their decades of experience in optimization to provide an enriching layer of historical context. Suitable for advanced undergraduates and masters students in management science, operations research, and related fields.” Michael P. Friedlander, IBM Professor of Computer Science, Professor of Mathematics, University of British Columbia
Linear and nonlinear optimization 1st Table of contents:
1. LP MODELS AND APPLICATIONS
1.1 A linear programming problem
1.2 Linear programs and linear functions
1.3 LP models and applications
1.4 Exercises
2. LINEAR EQUATIONS AND INEQUALITIES
2.1 Equivalent forms of the LP
2.2 Polyhedral sets
2.3 Exercises
3. THE SIMPLEX ALGORITHM
3.1 Preliminaries
3.2 A case where the sufficient condition is also necessary
3.3 Seeking another BFS
3.4 Pivoting
3.5 Steps of the Simplex Algorithm
3.6 Exercises
4. THE SIMPLEX ALGORITHM CONTINUED
4.1 Finding a feasible solution
4.2 Dealing with degeneracy
4.3 Revised Simplex Method
4.4 Exercises
5. DUALITY AND THE DUAL SIMPLEX ALGORITHM
5.1 Dual linear programs
5.2 Introduction to the Dual Simplex Algorithm
5.3 Exercises
6. POSTOPTIMALITY ANALYSES
6.1 Changing model dimensions
6.2 Ranging (Sensitivity analysis)
6.3 Parametric linear programming
6.4 Shadow prices and their interpretation
6.5 Exercises
7. SOME COMPUTATIONAL CONSIDERATIONS
7.1 Problems with explicitly bounded variables
7.2 Constructing a starting (feasible) basis
7.3 Steepest-edge rule for incoming column selection
7.4 Structured linear programs
7.5 Computational complexity of the Simplex Algorithm
7.6 Exercises
8. NLP MODELS AND APPLICATIONS
8.1 Nonlinear programming
8.2 Unconstrained nonlinear programs
8.3 Linearly constrained nonlinear programs
8.4 Quadratic programming
8.5 Nonlinearly constrained nonlinear programs
8.6 Exercises
9. UNCONSTRAINED OPTIMIZATION
9.1 Generic Optimization Algorithm
9.2 Optimality conditions for univariate minimization
9.3 Finite termination versus convergence of algorithms
9.4 Zero-finding methods
9.5 Univariate minimization
9.6 Optimality conditions for multivariate minimization
9.7 Methods for minimizing smooth unconstrained functions
9.8 Steplength algorithms
9.9 Exercises
10. DESCENT METHODS
10.1 The Steepest-descent Method
10.2 Newton’s Method
10.3 Quasi-Newton Methods
10.4 The Conjugate-gradient Method
10.5 Exercises
11. OPTIMALITY CONDITIONS
11.1 Statement of the problem
11.2 First-order optimality conditions
11.3 Second-order optimality conditions
11.4 Convex programs
11.5 Elementary duality theory for nonlinear programming
11.6 Exercises
12. PROBLEMS WITH LINEAR CONSTRAINTS
12.1 Linear equality constraints
12.2 Methods for computing
12.3 Linear inequality constraints
12.4 Active Set Methods
12.5 Special cases
12.6 Exercises
13. PROBLEMS WITH NONLINEAR CONSTRAINTS
13.1 Nonlinear equality constraints
13.2 Nonlinear inequality constraints
13.3 Overview of algorithm design
13.4 Penalty-function Methods
13.5 Reduced-gradient and Gradient-projection Methods
13.7 Projected Lagrangian Methods
13.8 Sequential Quadratic Programming (SQP) Methods
13.9 Exercises
14. INTERIOR-POINT METHODS
14.1 Barrier-function methods
14.2 Primal barrier-function method for linear programs
14.3 Primal-Dual barrier function for linear programs
14.4 Exercises
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