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Applied Optimization

Book Description

The starting point in the formulation of any numerical problem is to take an intuitive idea about the problem in question and to translate it into precise mathematical language. This book provides step-by-step descriptions of how to formulate numerical problems and develops techniques for solving them. A number of engineering case studies motivate the development of efficient algorithms that involve, in some cases, transformation of the problem from its initial formulation into a more tractable form. Five general problem classes are considered: linear systems of equations, non-linear systems of equations, unconstrained optimization, equality-constrained optimization and inequality-constrained optimization. The book contains many worked examples and homework exercises and is suitable for students of engineering or operations research taking courses in optimization. Supplementary material including solutions, lecture slides and appendices are available online at www.cambridge.org/9780521855648.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. List of Illustrations
  8. Preface
  9. 1. Introduction
    1. 1.1 Goals
    2. 1.2 Course plans
    3. 1.3 Model formulation and development
    4. 1.4 Overview
    5. 1.5 Pre-requisites
  10. 2. Problems, algorithms, and solutions
    1. 2.1 Decision vector
    2. 2.2 Simultaneous equations
    3. 2.3 Optimization
    4. 2.4 Algorithms
    5. 2.5 Solutions of simultaneous equations
    6. 2.6 Solutions of optimization problems
    7. 2.7 Sensitivity and large change analysis
    8. 2.8 Summary
  11. 3. Transformation of problems
    1. 3.1 Objective
    2. 3.2 Variables
    3. 3.3 Constraints
    4. 3.4 Duality
    5. 3.5 Summary
  12. Part I: Linear simultaneous equations
    1. 4. Case studies
      1. 4.1 Analysis of a direct current linear circuit
      2. 4.2 Control of a discrete-time linear system
    2. 5. Algorithms
      1. 5.1 Inversion of coefficient matrix
      2. 5.2 Solution of triangular systems
      3. 5.3 Solution of square, non-singular systems
      4. 5.4 Symmetric coefficient matrix
      5. 5.5 Sparsity techniques
      6. 5.6 Changes
      7. 5.7 Ill-conditioning
      8. 5.8 Non-square systems
      9. 5.9 Iterative methods
      10. 5.10 Summary
  13. Part II: Non-linear simultaneous equations
    1. 6. Case studies
      1. 6.1 Analysis of a non-linear direct current circuit
      2. 6.2 Analysis of an electric power system
    2. 7. Algorithms
      1. 7.1 Newton–Raphson method
      2. 7.2 Variations on the Newton–Raphson method
      3. 7.3 Local convergence of iterative methods
      4. 7.4 Globalization procedures
      5. 7.5 Sensitivity and large change analysis
      6. 7.6 Summary
    3. 8. Solution of the case studies
      1. 8.1 Analysis of a non-linear direct current circuit
      2. 8.2 Analysis of an electric power system
  14. Part III: Unconstrained optimization
    1. 9. Case studies
      1. 9.1 Multi-variate linear regression
      2. 9.2 Power system state estimation
    2. 10. Algorithms
      1. 10.1 Optimality conditions
      2. 10.2 Approaches to finding minimizers
      3. 10.3 Sensitivity
      4. 10.4 Summary
    3. 11. Solution of the case studies
      1. 11.1 Multi-variate linear regression
      2. 11.2 Power system state estimation
  15. Part IV: Equality-constrained optimization
    1. 12. Case studies
      1. 12.1 Least-cost production
      2. 12.2 Power system state estimation with zero injection buses
    2. 13. Algorithms for linear constraints
      1. 13.1 Optimality conditions
      2. 13.2 Convex problems
      3. 13.3 Approaches to finding minimizers
      4. 13.4 Sensitivity
      5. 13.5 Solution of the least-cost production case study
      6. 13.6 Summary
    3. 14. Algorithms for non-linear constraints
      1. 14.1 Geometry and analysis of constraints
      2. 14.2 Optimality conditions
      3. 14.3 Approaches to finding minimizers
      4. 14.4 Sensitivity
      5. 14.5 Solution of the zero injection bus case study
      6. 14.6 Summary
  16. Part V: Inequality-constrained optimization
    1. 15. Case studies
      1. 15.1 Least-cost production with capacity constraints
      2. 15.2 Optimal routing in a data communications network
      3. 15.3 Least absolute value estimation
      4. 15.4 Optimal margin pattern classification
      5. 15.5 Sizing of interconnects in integrated circuits
      6. 15.6 Optimal power flow
    2. 16. Algorithms for non-negativity constraints
      1. 16.1 Optimality conditions
      2. 16.2 Convex problems
      3. 16.3 Approaches to finding minimizers: active set method
      4. 16.4 Approaches to finding minimizers: interior point algorithm
      5. 16.5 Summary
    3. 17. Algorithms for linear constraints
      1. 17.1 Optimality conditions
      2. 17.2 Convex problems
      3. 17.3 Approaches to finding minimizers
      4. 17.4 Sensitivity
      5. 17.5 Summary
    4. 18. Solution of the linearly constrained case studies
      1. 18.1 Least-cost production with capacity constraints
      2. 18.2 Optimal routing in a data communications network
      3. 18.3 Least absolute value estimation
      4. 18.4 Optimal margin pattern classification
    5. 19. Algorithms for non-linear constraints
      1. 19.1 Geometry and analysis of constraints
      2. 19.2 Optimality conditions
      3. 19.3 Convex problems
      4. 19.4 Approaches to finding minimizers
      5. 19.5 Sensitivity
      6. 19.6 Summary
    6. 20. Solution of the non-linearly constrained case studies
      1. 20.1 Optimal margin pattern classification
      2. 20.2 Sizing of interconnects in integrated circuits
      3. 20.3 Optimal power flow
  17. References
  18. Index